Given $f (x) = {cos}^{2} x$ , find whether $f (x)$ is increasing or decreasing in the range $[0, \frac{π}{2}]$ .

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Solution

$f (x) = cos x$
differentiating w.r.t $x$
$f^{'} (x) = 2 cos x . (- sin x) = - sin 2 x$
here,
$\begin{matrix} 0 \leq x \leq \frac{π}{2} 0 \leq 2 x \leq π sin 2 x \geq 0 - sin 2 x \leq 0 f^{'} (x) \leq 0 \end{matrix}$
$f (x)$ is decreasing

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