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

## Chapter 18 - Maxima and Minima Exercise MCQ

Let the function be 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

Given: is 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.

If x is real, the minimum value of is

a. -1

b. 0

c. 1

d. 2

Let

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

The maximum value of is

a.

b.

c.

d.

Let

Differentiating w.r.t x, we get

Take f'(x) = 0, we get

cos 2x = 0

Again differentiating, we get

At

Therefore, is the point of maxima.

The maximum value of f(x) is

The maximum value of is

a. e

b.

c.

d.

Let

Taking log on both the sides, we get

… (i)

Differentiating w.r.t x, we get

Differentiating w.r.t x, we get

… (iii)

From equation (ii), we get

Take we get

At we have

Therefore, is the point of maxima.

The maximum value of f(x) is

The function has a stationary point at

a. x = e

b.

c. x = 1

d.

Let

Taking log on both the sides, we get

… (i)

Differentiating w.r.t x, we get

Take we get

Thus, is the stationary point.

Maximum slope of the curve is

a. 0

b. 12

c. 16

d. 32

Given:

Differentiating w.r.t x, we get

which is the slope of the curve

Differentiating w.r.t x, we get

Take

Again differentiating w.r.t x, we get

So, the slope is maximum at x = 1

The function has

a. two points of local maximum

b. two points of local minimum

c. one maximum and one minimum

d. no maximum no minimum

Given:

Differentiating w.r.t x, we get

Take f'(x) = 0, we get

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

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.

The maximum value of x^{1/x}, x > 0 is

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

(a) -2

(b) -1

(c) 2

(d) 4

## Chapter 18 - Maxima and Minima Exercise Ex. 18.1

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.

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

Find the maximum and minimum values, if any, usingwithout derivatives of the following function given by on R.

Find the maximum and minimum values, if any, without using derivatives of the following function given by on R.

## Chapter 18 - Maxima and Minima Exercise Ex. 18.2

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^{3 }- 3x

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)* = sin*x* - cos *x*, 0 < *x* < 2π

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:

## Chapter 18 - Maxima and Minima Exercise Ex. 18.3

Show that the function given by *f(x)*= has maximum value at *x = e*.

If *f(x)* = *x*^{3} + *ax*^{2} + *bx* + *c* has a maximum at
*x* = -1 and minimum at *x* = 3. Determine *a*, *b* and *c*.

Prove that has maximum value at

Given:

Differentiating w.r.t x, we get

Take f'(x) = 0

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

At

Clearly, f"(x) < 0 at

Thus, is the maxima.

Hence, f(x) has maximum value at .

## Chapter 18 - Maxima and Minima Exercise Ex. 18.4

Find both the absolute maximum and absolute minimum of 3x^{4} - 8x^{3} + 12x^{2} - 48x + 25 on the interval [0, 3]

## Chapter 18 - Maxima and Minima Exercise Ex. 18.5

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

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.

An isosceles triangle of vertical angle 2θ is inscribed in a circle radius a. show that the area of the triangle is maximum when

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)

Amongst
all open (from the top) right circular cylindrical boxes of volume 125π cm^{3},
find the dimensions of the box which has the least surface area.

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

Volume of cylinder

… (i)

Surface area of cylinder

From (i), we get

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is

Let be an isosceles triangle with AB = AC.

Let

Here, AO bisects

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

OE = OF = OD = r (radius)

Now,

In

Similarly, AF = r cot x

In

As OB bisect we have

In

Similarly, BD = DC = CE =

We have, perimeter of

P = AB + BC + CA

= AE + EC + BD + DC + AF + BF

Differentiating w.r.t x, we get

Taking

As

Therefore, is an equilateral triangle.

Taking second derivative of P, we get

At

Therefore, the perimeter is minimum when

Least value of P

## Chapter 18 - Maxima and Minima Exercise Ex. 18VSAQ

Write the maximum value of *f(x)* = x+, x < 0.

### Kindly Sign up for a personalised experience

- Ask Study Doubts
- Sample Papers
- Past Year Papers
- Textbook Solutions

#### Sign Up

#### Verify mobile number

Enter the OTP sent to your number

Change