Max Value:

The maximum of $\cos(\text{anything})$ is $1$ . At $x = 0$ , both summands attain their maximum, so the maximum of $f(x)$ is $2$ , and it is attained at at least one point.

To attain the maximum anywhere else, we must get both $\cos$ addends to have value $1$ , so we need $x \neq 0$ , and integers $m$ and $n$ such that:

$x = 2\pi n$

$\sqrt{2} x = 2\pi m$

Substitute, and we get:

$\sqrt{2} (2\pi n) = 2\pi m$

$\sqrt{2} = \frac{m}{n}$

Since $\sqrt{2}$ is irrational, and $\frac{m}{n}$ is rational, the system of equations has no solution. So we are stuck with the $\boxed{1}$ maxima at $x = 0$ as the only maxima.