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

Hint: 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 \[\dfrac{T}{{\left( {\dfrac{1}{n}} \right)}}\] \[ \Rightarrow nT\].