Question

The function $f\left( x \right) = \left| {\sin 4x} \right| + \left| {\cos 2x} \right|$, is a periodic function with period
A) $2\pi$
B) $\pi$
C) $\dfrac{\pi }{2}$
D) $\dfrac{\pi }{4}$

Answer 1

Here we first use the known fact that the period of $\sin x$ $\&$ $\cos x$ is $2\pi$.
Also, the period of every function in modulus is $\pi$ and also, if the period of function g(x) is T then the period of $g\left( {nx} \right)$ is $\dfrac{T}{n}$. We will find the periods of $\left| {\sin 4x} \right|$ and $\left| {\cos 2x} \right|$ separately and then take the LCM of numerators and gcd of denominators to find the final answer.

Complete step-by-step answer:
The given function is:-
$f\left( x \right) = \left| {\sin 4x} \right| + \left| {\cos 2x} \right|$
Here we will first find the period of $\left| {\sin 4x} \right|$.
Now we know that the period of $\sin x$ is $2\pi$
Also, the period of every function in modulus is $\pi$
Hence, the period of $\left| {\sin x} \right|$ is $\pi$.
Now we need to find the period of $\left| {\sin 4x} \right|$ and we know that if the period of function g(x) is T then the period of $g\left( {nx} \right)$ is $\dfrac{T}{n}$
Therefore, the period of $\left| {\sin 4x} \right|$ becomes $\dfrac{\pi }{4}$…………………………………(1)
Now we will find the period of $\left| {\cos 2x} \right|$.
Now we know that the period of $\cos x$ is $2\pi$
Also, the period of every function in modulus is $\pi$
Hence, the period of $\left| {\cos x} \right|$ is $\pi$.
Now we need to find the period of $\left| {\cos 2x} \right|$ and we know that if the period of function g(x) is T then the period of $g\left( {nx} \right)$ is $\dfrac{T}{n}$
Therefore, the period of $\left| {\cos 2x} \right|$ becomes $\dfrac{\pi }{2}$…………………………………(2)
From equations 1 and 2 we got:
Period of $\left| {\sin 4x} \right|$ is $\dfrac{\pi }{4}$.
Period of $\left| {\cos 2x} \right|$ is $\dfrac{\pi }{2}$.
Since we have to find the period of $f\left( x \right) = \left| {\sin 4x} \right| + \left| {\cos 2x} \right|$
Hence we have to find the LCM of numerators of periods of $\left| {\sin 4x} \right|$ and $\left| {\cos 2x} \right|$ i.e. we have to find the LCM of $\left( {\pi ,\pi } \right)$ as the numerator and the G.C.D of denominators of periods of $\left| {\sin 4x} \right|$ and $\left| {\cos 2x} \right|$ i.e. G.C.D of $\left( {4,2} \right)$ as the denominator.
Therefore, the LCM is $\pi$ and the G.C.D of $\left( {4,2} \right)$ is 2.
Hence the period of $f\left( x \right) = \left| {\sin 4x} \right| + \left| {\cos 2x} \right|$ is $\dfrac{\pi }{2}$

So, the correct answer is “Option C”.

Note: Students should have pre-knowledge to solve such questions like the period of $\sin x$ $\&$ $\cos x$ is $2\pi$.
Also, students should note that if the period of function g(x) is T then the period of $g\left( {nx} \right)$ is $\dfrac{T}{n}$ and the period of $g\left( {\dfrac{x}{n}} \right)$ is $\dfrac{T}{{\left( {\dfrac{1}{n}} \right)}}$ $\Rightarrow nT$.