Let x, y, z be real numbers with x greater or equal than y greater or equal than zgreater or equal than straight pi over 12 such that x+y+z= straight pi over 2 and P = cosx.siny.cosz. then what are the minimum and maximum values of P? Also mention at what values of x,y,z they occur? 

Asked by sarn1027 | 5th May, 2020, 02:44: PM

Expert Answer:

Given:
 x greater or equal than y greater or equal than zgreater or equal than straight pi over 12 ... (i)
x+y+z= straight pi over 2 ... (ii)
 
P = cosx siny cosz
rightwards double arrow P equals fraction numerator cos z over denominator 2 end fraction open parentheses 2 space cos x space sin y close parentheses
rightwards double arrow P equals fraction numerator cos z over denominator 2 end fraction open square brackets sin open parentheses x plus y close parentheses minus sin open parentheses x minus y close parentheses close square brackets less or equal than fraction numerator cos z over denominator 2 end fraction space sin open parentheses x plus y close parentheses
rightwards double arrow P less or equal than fraction numerator cos z over denominator 2 end fraction space sin open parentheses straight pi over 2 minus z close parentheses space space... space U sin g space left parenthesis i i right parenthesis
rightwards double arrow P less or equal than fraction numerator cos z over denominator 2 end fraction space cos z
rightwards double arrow P less or equal than fraction numerator cos squared z over denominator 2 end fraction equals 1 half cross times fraction numerator open parentheses 1 plus cos 2 z close parentheses over denominator 2 end fraction equals fraction numerator open parentheses 1 plus cos 2 z close parentheses over denominator 4 end fraction
rightwards double arrow P less or equal than 1 fourth open parentheses 1 plus cos straight pi over 6 close parentheses space space... space U sin g space left parenthesis i right parenthesis
rightwards double arrow P less or equal than 1 fourth open parentheses 1 plus fraction numerator square root of 3 over denominator 2 end fraction close parentheses
rightwards double arrow P less or equal than fraction numerator square root of 3 plus 2 over denominator 8 end fraction

Answered by Renu Varma | 6th May, 2020, 11:33: AM

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