<s>[INST] How to find the absolute maximum of $f(x) = (\sin 2\theta)^2 (1+\cos 2\theta)$ for $0 \le \theta \le \frac{\pi}2$? How to find the absolute maximum of $f(x) = (\sin 2\theta)^2 (1+\cos 2\theta)$ for $0 \le \theta \le \frac{\pi}2$?   
I found the derivative to be $f'(x)= (2\sin 4\theta)(1+\cos 2\theta) + (\sin 2\theta)^2 (-2\sin 2\theta)$.  
Now I know that I have to set this equal to zero and solve for $\theta$ but that is where I get stuck, can I solve for $\theta$ or is there a different way that is easier?
 [/INST] In order to find the stationary points of $g^2$ it is enough to find the roots of $g$ and the stationary points of $g$. Since:
$$ \sin^2(2\theta)(1+\cos(2\theta)) = 2\sin^2(2\theta)\cos^2(\theta) $$
we just need to find the stationary points of:
$$ 2\sin(2\theta)\cos(\theta) = \sin(3\theta)+\sin(\theta) $$
that occur for $\cos\theta = 0$ or $3\cos^2\theta-2=0$. If $\cos^2\theta=\frac{2}{3}$, $\sin^2\theta=\frac{1}{3}$. That gives:
$$ \sin^2(2\theta)(1+\cos(2\theta))\leq 8\cdot\frac{1}{3}\cdot\frac{4}{9} = \frac{32}{27}.$$
That also follows from the AM-GM inequality.
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