Question

The period of the function $f\left(x\right)=\left|\cos \left(\frac{x}{2}\right)\right|+\left|\sin x\right|$ is:

• A. $2\mathrm{\pi }$
• B. $\mathrm{\pi }$
• C. $4\mathrm{\pi }$
• D. None of these

Answer 1

The correct option is A

$2\mathrm{\pi }$

Explanation for the correct option:

Compute the period of the given function.

We know that the period of $\sin \left(x\right)=2\pi$ and $\left|\sin \left(x\right)\right|=\pi$

And the period of $\cos \left(x\right)=2\pi$ and $\left|\cos \left(x\right)\right|=\pi$

So, period of $\left|\cos \left(\frac{x}{2}\right)\right|=2\pi$

Now, given $f\left(x\right)=\left|\cos \left(\frac{x}{2}\right)\right|+\left|\sin x\right|$

Period of $f\left(x\right)=g\left(x\right)+h\left(x\right)=$ L.C.M $(T_1,T_2)$

$\left|\cos \left(\frac{x}{2}\right)\right|=2\pi$

$\left|\sin \left(x\right)\right|=\pi$

Thus, the Period of $f\left(x\right)$ $=2\pi$

Hence, Option$\left(\mathrm{A}\right)$ is the correct answer