Question

Let a function $f\left(x\right)=\cos 2x+2\sin x$ is defined in $[0,\pi ],$ then which of the following is not correct

• A. global maxima exist at $x=\frac{\pi }{6}$
• B. global maxima exist at $x=\frac{5\pi }{6}$
• C. global minima exist at $x=\frac{\pi }{2}$
• D. global extrema does not exist.

Answer 1

The correct option is D global extrema does not exist.

Given : $f\left(x\right)=\cos 2x+2\sin x$
$\Rightarrow f^{\prime }\left(x\right)=-2\sin 2x+2\cos x\Rightarrow f^{\prime }\left(x\right)=2\cos x(1-2\sin x)$
for critical points : $f^{\prime }\left(x\right)=0$
$\Rightarrow 2\cos x(1-2\sin x)=0\Rightarrow x=\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6}$ and end points are $0,\pi .$
Now, value of function at these points :
$f\left(0\right)=1,f\left(\frac{\pi }{6}\right)=\frac{3}{2},f\left(\frac{\pi }{2}\right)=1,f\left(\frac{5\pi }{6}\right)=\frac{3}{2},f\left(\pi \right)=1$
So, range of $f\left(x\right)\in [1,\frac{3}{2}]$
and global maxima exist at $x=\frac{\pi }{6},\frac{5\pi }{6}$ and
global minima exist at $x=0,\frac{\pi }{2},\pi$