Find maximum/minimum for $\cos(2x) + \cos(y) + \cos(2x+y) $ I have not been able to find the critical points for $\cos(2x) + \cos(y) + \cos(2x+y) $

When you set the partial derivatives equal to $0$, you obtain the two equations
\begin{align*}
\sin 2x + \sin (2x+y) &= 0 \\
\sin y + \sin (2x+y) &= 0\,.
\end{align*}
This means that we must have $\sin 2x = \sin y$, and so either $y=x+(2k)\pi$ or $y=(2k+1)\pi-x$ for some integer $k$. Substituting, for example, $y=2x$, you'll find that
$\sin 2x = 0$ or $\cos 2x = -1/2$. Continue :)
