The number of values of x where the function $f(x)=\cos x+\cos (\sqrt{2} x)$ attains its maximum is

1

$\cos (\sqrt{2} x)$ has period $\frac{2 \pi}{\sqrt{2}}=\sqrt{2} x$ and cos x has period $2 \pi$.

Hence, after attaining a maximum value at x = 0 (i.e. x = 0, y = 2), the function $f(x)=$ $\cos x+\cos (\sqrt{2} x)$ will not attain this value again.