InterviewSolution
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InterviewSolution
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In ` A B C`Prove that `cos^2A/2+cos^2B/2+cos^2C/2lt=9/4dot`In `cos^2A/2+cos^2B/2+cos^2C/2=y(x^2+1/(x^2))`then find the maximum value of `ydot` |
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Answer» Correct Answer - `9//8` (a) In `triangle ABC`, we know that `cosA+cosB+cos Cle(3)/(2)` `cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)` `=(1+cosA)/(2)+(1+cosB)/(2)+(1+cosC)/(2)` `=(3)/(2)+(cos A+cosB+cosC)/(2)le(3)/(2)+(3)/(4)` (using eq. i) `therefore cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)le(9)/(4)`. (b) `cos^(2)""(A)/(2)+cos^(2)""(B)/(2)+cos^(2)""(C)/(2)=y(x^(2)+(1)/(x^(2)))` `therefore y(x^(2)+(1)/(x^(2)))le(9)/(4)`. `therefore yle(9)/(4(x^(2)+(1)/(x^(2)))` Now `x^(2)+(1)/(x^(2))ge2` `therefore yle(9)/(8)`. thus, maximum value of `y` is `(9)/(8)`. |
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