Question

Complete set of values of '$m$' for which function $f\left(x\right)=e^{\cos x}+2m\cos x+1$ is increasing $\forall xϵ(0,\frac{\pi }{2})$ is

• A. $(-\omega ,-\frac{1}{2})$
• B. $(-\frac{1}{2},\infty )$
• C. $(-\infty ,-\frac{e}{2})$
• D. $[-\frac{e}{2},\infty )$

Answer 1

The correct option is C $(-\infty ,-\frac{e}{2})$

$f^{\prime }\left(x\right)=e^{\cos x}(-\sin x)-2m\sin x$
$=-\sin x(e^{\cos x}+2m)\ge 0$

So $e^{\cos x}+2m\le 0[\because xϵ(0,\frac{\pi }{2}\left)\right]$
$m\le -\frac{e^{\cos x}}{2}$
$\therefore m<\frac{-e}{2}$
(maximum value of $\frac{-e^{\cos x}}{2}$)