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Let x, y, z be real numbers with $x greater or equal than y greater or equal than z$ $greater or equal than straight pi over 12$ 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 | 05 May, 2020, 02:44: PM

Expert Answer

Given:

x greater or equal than y greater or equal than z

greater 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 | 06 May, 2020, 11:33: AM

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