Question

The function $f(x)={\cos }^{2}x$ is strictly decreasing on

• A. $\left[0,\frac{\pi }{2}\right]$
• B. $\left[0,\frac{\pi }{2}\right)$
• C. $\left(0,\frac{\pi }{2}\right)$
• D. $\left(0,\frac{\pi }{2}\right]$

Answer 1

Answer: (C)

We have, $f(x)={\cos }^{2}x$
On differentiating both sides w.r.t. ‘$x$’, we get
$f^{\prime }(x)=-2\cos x\sin x=-\sin 2x$
$\because -1\le \sin 2x\le 1$
$\therefore f^{\prime }(x)>0$ for $x\in (\frac{-\pi }{2},0)$
and $f^{\prime }(x)<0$ for $x\in (0,\frac{\pi }{2})$
$\therefore f(x)={\cos }^{2}x$ is strictly decreasing on $(0,\frac{\pi }{2})$.