The function `f(x) = 3^(sin^(2)pi + sin^(4) pi x + x-[x])` where [x] denotes the greatest interger less than or equal to x, is

Options

A. a periodic function with period 1
B. a periodic function with period 2
C. a periodic function with period ` (1)/(2)`
D. not a periodic function

Answer

A

Explanation

We have
`sin^(2) pi x + sin^(4) pi x`
`=(1)/(2) {1-cos 2 pi x} +(1)/(8) {3-4 cos 2 pi x + cos 4 pi x}`
`=(7)/(8) - cos 2 pi x +(1)/(8) cos 4 pi x`
Clearly, it is a peridic function equal to LCMof
`(2pi)/(2pi)` and `(2pi)/(4)` i.e., 1.
Also, `x - [x]` is periodic with period 1.
Hence, `f(x)` is periodic with period 1.