RD SHARMA Solutions for Class 12-science Maths Chapter 18 - Maxima and Minima

Chapter 18 - Maxima and Minima Exercise MCQ

Question 30

Let the function Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minimabe defined by f(x) = 2x + cos x, then f(x)

a. has a minimum at x = π 

b. has a maximum at x = 0

c. is a decreasing function

d. is an increasing function

Solution 30

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minimais defined as f(x) = 2x + cos x

Differentiating w.r.t x, we get

f'(x) = 2 - sin x

As -sin x -1

Therefore, 1 - sin x > 0 for all x.

Hence, f(x) is an increasing function.

Question 31

If x is real, the minimum value of Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is

a. -1

b. 0

c. 1

d. 2

Solution 31

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Differentiating w.r.t x, we get

f'(x) = 2x - 8

Take f'(x) = 0, we get

2x - 8 = 0

i.e. x = 4

Again differentiating, we get

f"(x) = 2 > 0 for all real x

Therefore, x = 4 is the point of minima.

The minimum value of f(x) is

f(4) = 16 - 32 + 17 = 1

Question 32

The maximum value of Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is

a. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

b. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

c. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

d. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Solution 32

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Take f'(x) = 0, we get

cos 2x = 0

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Again differentiating, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

At Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And MinimaRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Therefore, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is the point of maxima.

The maximum value of f(x) is

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Question 33

The maximum value of Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is

a. e

b. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

c. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

d. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Solution 33

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Taking log on both the sides, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima … (i)

Differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima … (iii)

From equation (ii), we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Take Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

At Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima we have

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Therefore, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is the point of maxima.

The maximum value of f(x) is

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Question 34

The function Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima has a stationary point at

a. x = e

b. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

c. x = 1

d. Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Solution 34

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Taking log on both the sides, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima … (i)

Differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Take Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Thus, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is the stationary point.

Question 35

Maximum slope of the curve Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is

a. 0

b. 12

c. 16

d. 32

Solution 35

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima which is the slope of the curve

Differentiating Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Take Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Again differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

So, the slope is maximum at x = 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Question 36

The function Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima has

a. two points of local maximum

b. two points of local minimum

c. one maximum and one minimum

d. no maximum no minimum

Solution 36

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Take f'(x) = 0, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Now, let's find f(x) at x = 2 or -1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Therefore, x = -1 is the point of local maxima and the maximum value is 11.

Whereas, x = 2 is the point of local minima and the minimum value is -16.

Hence, f(x) has one maximum and one minimum.

Question 1

The maximum value of x1/x, x > 0 is

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 3

 Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 27

Let f(x) = 2x3 - 3x2 - 12x + 5 on [-2,4]. The relative maximum occurs at x = 

(a) -2

(b) -1 

(c) 2

(d) 4

Solution 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Chapter 18 - Maxima and Minima Exercise Ex. 18.1

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 2

Find the maximum and minimum values, if any, without using derivatives of the following function given by f(x) = -(x-1)2 + 2 on R.

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 4

Find the maximum and minimum values, if any, without using derivatives of the following function given by h(x) = sin(2x) + 5 on R.

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 5

Find the maximum and minimum values, if any,  usingwithout derivatives of the following function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima on R.

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 7

Find the maximum and minimum values, if any, without using derivatives of the following function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima on R.

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Chapter 18 - Maxima and Minima Exercise Ex. 18.2

Question 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 2

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

f(x) =x- 3x

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 8

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

f(x) = sinx - cos x, 0 < x < 2π

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 12

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Chapter 18 - Maxima and Minima Exercise Ex. 18.3

Question 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(v)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(vi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(vii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(vii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(viii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 1(viii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 1(ix)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 1(ix)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 1(x)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(x)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(xi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(xi)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(xii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(xii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 2(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 2(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 2(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 2(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 2(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 2(iii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 4

Show that the function given by f(x)=Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima has maximum value at x = e.

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 7

If f(x) = x3 + ax2 + bx + c has a maximum at x = -1 and minimum at x = 3. Determine a, b and c.

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 8

Prove that Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima has maximum value at Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 8

Given: Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Take f'(x) = 0

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Differentiating f'(x) w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

At Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Clearly, f"(x) < 0 at Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Thus, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is the maxima.

Hence, f(x) has maximum value at Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima.

Chapter 18 - Maxima and Minima Exercise Ex. 18.4

Question 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(i)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(ii)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 1(iii)

Find both the absolute maximum and absolute minimum of 3x4 - 8x3 + 12x2 - 48x + 25 on the interval [0, 3] 

Solution 1(iii)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1(iv)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Chapter 18 - Maxima and Minima Exercise Ex. 18.5

Question 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 2
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 3
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 4

Divide 15 into two parts such that the square of one multiplied with the cube of the other is maximum.

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 12

A square piece of tin of side 18 cm is to made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible? Also, find this maximum volume.

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And MinimaRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 14
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 14
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 16
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 17
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 17
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 18
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 18
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 19
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 19
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And MinimaRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 20
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Solution 20
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 22

An isosceles triangle of vertical angle 2θ is inscribed in a circle radius a. show that the area of the triangle is maximum when Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 22

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And MinimaRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

 

Question 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 37

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 37

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 38

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 38

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 39

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 39

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 40

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 40

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 41

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its semi - circular ends is π : ( π + 2) 

Solution 41

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 42

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 42

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 43

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 43

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 44

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 44

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima
Question 45

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 45

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 45

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 45

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 5

Amongst all open (from the top) right circular cylindrical boxes of volume 125π cm3, find the dimensions of the box which has the least surface area.

Solution 5

Let r and h be the radius and height of the cylinder.

Volume of cylinder Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima … (i)

Surface area of cylinder Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

From (i), we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Question 23

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Solution 23

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima be an isosceles triangle with AB = AC.

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Let Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Here, AO bisects Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Taking O as the centre of the circle, join OE, OF and OD such that

OE = OF = OD = r (radius)

Now, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

In Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Similarly, AF = r cot x

In Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

As OB bisect Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima we have

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

In Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Similarly, BD = DC = CE = Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

We have, perimeter of Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

P = AB + BC + CA

 = AE + EC + BD + DC + AF + BF

 Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Differentiating w.r.t x, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Taking Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

As Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Therefore, Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima is an equilateral triangle.

Taking second derivative of P, we get

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

At Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Therefore, the perimeter is minimum when Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Least value of P Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

 Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

 Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima 

Chapter 18 - Maxima and Minima Exercise Ex. 18VSAQ

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 5

Write the maximum value of f(x) = x+Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima, x < 0. 

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And MinimaRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Maxima And Minima