Question

Find the greatest and the least values of the following functions.
Fin the extrema of the function $f\left(x\right)=2x\sin 2x+\cos 2x-\sqrt{3}$ on the interval $[-\pi /2,3\pi /8].$

Answer 1

$f\left(x\right)=2x\sin 2x+\cos 2x-\sqrt{3}$

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

$x=0,\frac{(2n+1)\pi }{4}$

$f^{\prime \prime }\left(x\right)=4\cos 2x-8x\sin 2x$

$f^{\prime \prime }\left(0\right)=4>0$ Hence $f\left(0\right)=-\sqrt{3}-1$ is minimum

$f^{\prime \prime }\left(\frac{\pi }{4}\right)=-2\pi <0$ Hence $f\left(\frac{\pi }{4}\right)=\frac{\pi }{2}-\sqrt{3}$ is maximum