The function f(x)= ( 2 )+2 ( 3 ) - ( 4 ) +2 ( 3 )- ( 4 ) is perio… - Vidyadip

MCQ

The function $ \text{f}(\text{x})=\sin(\frac{\pi\text{x}}{2})+2\cos\Big(\frac{\pi\text{x}}{3}\Big) -\tan\Big(\frac{\pi\text{x}}{4}\Big) +2\cos\Big(\frac{\pi\text{x}}{3}\Big)-\tan\Big(\frac{\pi\text{x}}{4}\Big)$ is periodic with period:

A
$4$
B
$6$
C
$8$
✓
$12$

✓

Answer

Correct option: D.

$12$

Period of $\sin \sin\Big(\frac{\pi\text{x}}{2}\Big)=\frac{2\pi}{\big(\frac{\pi\text{x}}{2}\big) }=4=\frac{2\pi}{\big(\frac{\pi\text{x}}{2}\big)}=4$
Period of $\cos \cos\Big(\frac{\pi\text{x}}{3}\Big)=\frac{2\pi}{\big(\frac{\pi\text{x}}{3}\big) }=6 $
Period of $\tan \sin\Big(\frac{\pi\text{x}}{2}\Big)=\frac{2\pi}{\big(\frac{\pi\text{x}}{2}\big) }=4$
So, period of $f(x) = \text{LCM} (4, 6, 4) = 12$

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