JEE Main 2014 — Functions Question with Solution

Given $f\left(x\right)=\left|\sin 4x\right|+\left|\cos 2x\right|$

We know that the period of $\left|sinx\right|$ is $\pi$ and also if $f\left(x\right)$ is a periodic function with fundamental period $T,$ then the fundamental period of $f\left(kx\right)$ is $\frac{T}{\left|k\right|}.$

$\Rightarrow T_1=$period of $\left|\text{sin4}\mathit{\text{x}}\right|=\frac{\pi }{4}$

Also, we know that the period of $\left|\cos x\right|$ is $\pi$

$\Rightarrow T_2=$Period of $\left|\cos 2x\right|=\frac{\pi }{2}$

Again, we know that the period of the sum or difference of two periodic functions is the $L.C.M.$ of their periods.

So, the period of $f\left(x\right)$ is $L.C.M.$ of the periods of $\left|\sin 4x\right|$ and $\left|\cos 2x\right|$ i.e. the $L.C.M. \left(T_1, T_2\right)$

Now, $L.C.M. \left(\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{2}\right)=\frac{L.C.M. \left(\mathrm{\pi }, \mathrm{\pi }\right)}{H.C.F. \left(4, 2\right)}=\frac{\mathrm{\pi }}{2}.$