What is the maximum value of 2 x · 2 x ?

$$\begin{aligned} & y=\sin 2 x \cdot \cos 2 x \\ & \frac{d y}{d x}=\frac{d}{d x}(\sin 2 x) \cdot \cos 2 x+\frac{d}{d x}(\cos 2 x) \cdot \sin 2 x\end{aligned}$$=\cos ^2 2 x-\sin ^2 2 x=\cos 4 x$For maximum or minimum value$\frac{d y}{d x}=0 \Rightarrow \cos 4 x=0\therefore 4 x=\frac{\pi}{2} \Rightarrow x=\frac{\pi}{2 \times 4}$Now,$\frac{d^2 y}{d x^2}=-4 \cdot \sin 4 x$At$x=\frac{\pi}{2 \times 4}, \frac{d^2 y}{d x^2}=-4 \cdot \sin \frac{\pi}{2}=-4$.$\therefore y$is max. at$x=\frac{\pi}{2 \times 4}$$\begin{aligned} y_{\max } & =\sin 2\left(\frac{\pi}{2 \times 4}\right) \cdot \cos 2\left(\frac{\pi}{2 \times 4}\right) \\ & =\sin \frac{\pi}{4} \cdot \cos \frac{\pi}{4}=\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}=\frac{1}{2}\end{aligned}$$$