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# Find the critical numbers and stationary points of each of the following functions. $\;f(\theta)=\sin^{2} 2\theta \;$in$\;[0 , \pi ]$

Note: This is part 5 of a 6 part question, split as 6 separate questions here.

Toolbox:
• A ciritical number of a function f is a number c in the domain of f such that either $f'(c)=0$ or $f'(c)$ does not exist.
• Stationary points correspond to critical numbers for which $f'(c)=0$
$f(0)=\sin ^2 2 \theta$ on $[0,2 \pi]$
Step 1:
$f'(\theta)=2 \sin 2 \theta \cos 2 \theta.2$
$\qquad= 4 \sin 2 \theta \cos 2 \theta$
$\qquad=2 \sin 4 \theta$
$f'(0)=0=>2 \sin 4 \theta=0$
$4 \theta=n \pi=>n =z$
$\theta= \large\frac{n \pi}{4} $$n \in z Step 2: The values of \theta in [0, \pi] are 0, \large\frac{\pi}{4},\frac{\pi}{2}, \frac{3 \pi}{4}$$,\pi$ (critical number)
$x=0=>f(0)=0;$
$x=\large\frac{\pi}{4}$$=>f(\large\frac{\pi}{4})$$=1;$
$x=\large\frac{\pi}{2}$$=>f(\large\frac{\pi}{2})$$=0;$
$x=\large\frac{3\pi}{4}$$=>f(\large\frac{3\pi}{4})$$=1;$
$x=\pi=>f(\pi)=0$
The critical points are $(0,0), (\large\frac{\pi}{4}$$,1), (\large\frac{\pi}{2},$$0),(\large\frac{3 \pi}{4}$$,1),(\pi,0)$