Period of a given function

Question

What is the period of the function $g(x)=|\cos(x)|+|\sin(x)|?$ This is a multiple choice question, and the value of $x=\pi/2$ satisfies the condition. But the period should be the smallest quantity. I want to check the value of $\pi/4.$ I do not understand how to proceed with the mods.

Answer 1

Suppose there is a period $p\in (0,\frac {\pi} 2)$. Note that $g(0)=1$. So we get $1=g(p)=|\cos p| +| \sin p|=\cos p + \sin p$. This gives $2\sin (\frac p 2)\cos (\frac p 2)=\sin p =1-\cos p=2\sin^{2} (\frac p 2)$ so either $\sin (\frac p 2)=0$ or $\cos (\frac p 2)=1$. But both these are false, so there is no smaller period than $\frac {\pi} 2$.