Question

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

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

Answer 1

Answer: (D)

Infinite

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

$\Rightarrow f^{\prime }\left(x\right)=-\sin x-\sqrt{2}\sin \sqrt{2}x$

$\Rightarrow f^{\prime \prime }\left(x\right)=-\cos x-2\cos \sqrt{2}x$

For extremum value, put $f^{\prime }\left(x\right)=0$

$\Rightarrow -\sin x-\sqrt{2}\sin \sqrt{2}x=0$

the above eqution can be satified by infinite number of values of x

At $\Rightarrow x=0,\pi ,2\pi ,3\pi ......f^{\prime \prime }\left(x\right)<0$ (maximum)

hence option D is correct.