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Question

f(x)=sin x (1+cos x), x(0,π2)Then, f(x) has a maxima at .

A
π2
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B
π6
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C
π4
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D
π3
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Solution

The correct option is D π3
f(x)=sin x (1+cos x), x(0,π2)f'(x)=cos x (1+cos x)sin2 xf'(x)=cos x+cos2 xsin2 xf'(x)=cos x+cos 2xand f''(x)=sin x 2 sin 2xFor maximum or minimum value of f(x),f'(x)=0cos x+cos 2x=0cos x=cos 2xcos x=cos (π±2x) x=(π±2x)x=π3Now, f''(π3)=2 sin 2π3sin π3 =23232=ve.Hence, f(x) has a maxima at x=π3.

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