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

1

We have,

$f(x)=\cos x+\cos \sqrt{2} x$

$\Rightarrow f(x)=2 \cos \left(\frac{\sqrt{2}+1}{2}\right) x \cos \left(\frac{\sqrt{2}-1}{2}\right) x \leq 2$

Clearly, $f(x)=2$ only when

$\cos \left(\frac{\sqrt{2}+1}{2}\right) x=1 \text { and } \cos \left(\frac{\sqrt{2}-1}{2}\right) x=1 \Rightarrow x=0$

Hence, there is only one value of x where f(x) attains its maximum.