Question

Find the greatest and the least values of the following functions.
$f\left(x\right)=sinx+cos2x$ on the interval $[0,\pi ].$

Answer 1

$f\left(x\right)=\sin x+\cos 2x$

$f^{\prime }\left(x\right)=\cos x-2\sin 2x=0\Rightarrow \cos x=4\sin x\cos x\Rightarrow x=\frac{\pi }{2},{\sin }^{-1}\frac{1}{4}$

$f^{\prime \prime }\left(x\right)=-\sin x-4\cos 2x$

$f^{\prime \prime }\left(\frac{\pi }{2}\right)=3>0$ Hence $f\left(\frac{\pi }{2}\right)=0$ is minimum value here

$f^{\prime \prime }\left({\sin }^{-1}\frac{1}{4}\right)=-\frac{26}{8}<0Hencef\left({\sin }^{-1}\frac{1}{4}\right)=\frac{9}{8}$ is maximum value