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Which one of the following is correct in respect of the function
$f(x)=x \sin x+\cos x+\frac{1}{2} \cos ^{2} x ?$
Options:
$f(x)=x \sin x+\cos x+\frac{1}{2} \cos ^{2} x ?$
Solution:
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Verified Answer
The correct answer is:
It is increasing in the interval $\left(0, \frac{\pi}{2}\right)$
$f(x)=x \sin x+\cos x+\frac{1}{2} \cos ^{2} x$
$\Rightarrow \quad f^{\prime}(x)=x \cos x+\sin x-\sin x-\sin x \cos x$
$=\cos x(x-\sin x)>0$ in $\left(0, \frac{\pi}{2}\right)$
$\Rightarrow \quad f^{\prime}(x)=x \cos x+\sin x-\sin x-\sin x \cos x$
$=\cos x(x-\sin x)>0$ in $\left(0, \frac{\pi}{2}\right)$
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