Question

Let $f\left(x\right)=\cos \sqrt{x}$, then which of the following is true?

• A. f(x) is periodic with period $\sqrt{2\pi }$
• B. f(x) is periodic with period $\sqrt{\pi }$
• C. f(x) is periodic with period 4${\pi }^2$
• D. f(x) is not a periodic function

Answer 1

The correct option is D f(x) is not a periodic function

$f\left(x\right)=\cos \sqrt{x}$ is not periodic which can be proved as follows
Let $\cos \sqrt{x}$ be periodic with period T, then $\cos \sqrt{x+T}=\cos \sqrt{x}$ or $\sqrt{x+T}=2n\pi \pm \sqrt{x}$
Squaring we get,
$x+T=4n^2{\pi }^2\pm 4n\pi \sqrt{x}+x$
or $T=4n^2{\pi }^2\pm 4n\pi \sqrt{x}$
Since T depends on x, therefore T can't be period. Therefore, cos $\sqrt{x}$ is not periodic.