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Question: The function \(f(x) = \sin\frac{\pi x}{2} + 2\cos\frac{\pi x}{3} - \tan\frac{\pi x}{4}\)is periodic ...

The function f(x)=sin⁡πx2+2cos⁡πx3−tan⁡πx4f(x) = \sin\frac{\pi x}{2} + 2\cos\frac{\pi x}{3} - \tan\frac{\pi x}{4}is periodic with

Period

A

6

B

3

C

4

D

12

Answer

12

Explanation

Solution

Period of sin⁡πx2=2ππ/2=4\sin\frac{\pi x}{2} = \frac{2\pi}{\pi/2} = 4

Period of cos⁡πx3=2ππ/3=6\cos\frac{\pi x}{3} = \frac{2\pi}{\pi/3} = 6 and period of tan⁡πx4=ππ/4=4\tan\frac{\pi x}{4} = \frac{\pi}{\pi/4} = 4

∴\therefore Period of f(x)=f(x) = L.C.M. of (4, 6, 4)=12.