Question

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

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is B

$1$

The explanation for the correct answer.

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

$\cos x$ has maximum value $1$ when $x=2n\mathrm{\pi }$ where $n=0,1,2,3,...$

$\cos \sqrt{2}x$ has maximum value $1$ when $x=\sqrt{2}m\mathrm{\pi }$ where $m=0,1,2,3,...$

Thus $f\left(x\right)=\cos x+\cos \left(\sqrt{2}x\right)$ has maximum value $2$ when both $\cos x$ and $\cos \sqrt{2}x$ are $1$

This is only possible when $x=0$

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

There exists no other value of $x$ for which the same value can be attained.

Hence, option B is the correct answer.