I would like to find the extrema of this function: $f(x) = w \cos(x) - (1-w)\cos(\alpha x)$. You can assume $w \in (0, 1)$ and $\alpha \in (0, 1)$.
Alternatively one could find the zeros of this function: $f'(x) = -w\sin(x)+\alpha(1-w)\sin(\alpha x)$.
There are ways to solve this numerically, but is there a (simple) analytical formula for the solution (e.g., depending on $w$ and $\alpha$)?