First slide

Extreme values and Periodicity of Trigonometric functions

Question

The function $f (x) = 3^{\sin^{2} π x + x - [x] + \sin^{4} π x}$ where $[x]$ denotes the greatest integer less than or equal to $x$ is

Moderate

A

a periodic function with period 1
B

a periodic function with period 2
C

a periodic function with period $\frac{1}{2}$
D

not a periodic function

Solution

We have $\sin^{2} π x + \sin^{4} π x$
$= \frac{1}{2} (1 - \cos 2 π x) + \frac{1}{8}$
$(3 - 4 \cos 2 π x + \cos 4 π x)$
$= \frac{7}{8} - \cos 2 π x + \frac{1}{8} \cos 4 π x$
Period = L.C.M of $\{\frac{2 π}{2 π}, \frac{2 π}{4 π}\} = 1$
also $x - [x]$ is periodic with period 1
$∴$ Period = 1

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