Multivariable function with cosines and a constraint

Question

What is the most negative value of $w=\cos{x} \cos{y} \cos(x+y)$ when $0\leq x,y < \frac{\pi}{2}$
What is the easiest way to approach this problem?
My attempt:
Finding partial derivatives and setting them to 0.

Answer 1

Hint

The problem being totally symmetric in $(x,y)$, just consider $$w=\cos ^2(x) \cos (2 x)\implies w'=-\sin (2 x)-\sin (4 x)=-2 \sin(3x)\cos(x)$$