<s>[INST] Maximum of |sin x| + |sin y| + |sin z| If $x$, $y$ and $z$ are real numbers with the property $x+y+z= \pi$, then the maximum of $\sin x+\sin y+\sin z$ is $3\sqrt{3}/2$.
Now, if $x+y+z=0$ then is the maximum of $|\sin x| + |\sin y| + |\sin z|$ again $3\sqrt{3}/2$?
 [/INST] $$f(x,y):=|\sin x|+|\sin y|+|\sin(-x-y)|=|\sin x|+|\sin y|+|\sin(x+y)|$$
In order to maximize $f$, let's consider $$\nabla f=\left\langle\frac{\sin x}{|\sin x|}\cos x+\frac{\sin (x+y)}{|\sin(x+y)|}\cos(x+y),\frac{\sin y}{|\sin y|}\cos y+\frac{\sin (x+y)}{|\sin(x+y)|}\cos(x+y)\right\rangle$$
If we require $\cos x-\cos(x+y)=0$, then $y=2k\pi-2x$ for some $k\in \Bbb Z$.  If their sum is equal to $0$, $y=(2k+1)\pi-2x$.
Then 
$$
\begin{align}
f(x)&=\left\{\begin{array}{l}|\sin x|+|\sin(2k\pi-2x)|+|\sin(2k\pi-x)|\\|\sin x|+|\sin((2k+1)\pi-2x)|+|\sin((2k+1)\pi-x)|\end{array}\right.\\
&=2|\sin x|+|\sin 2x|
\end{align}$$
Now, when is $f(x)$ maximized?  Well, $f'(x)=2\left(\frac{\sin x}{|\sin x|}\cos x+\frac{\sin 2x}{|\sin 2x|}\cos 2x\right)$.  So we either want $\cos x+\cos 2x=0$ or $\cos x-\cos 2x=0$  This gets $\cos x=\pm1,\pm\frac12$.  When this is true, we get:
$$f(x)=0,\frac{3\sqrt3}2$$
</s>