Question

The fundamental period of the function $f\left(x\right)=sin\left(\frac{\pi }{4}\right[x\left]\right)+tan\left(\frac{\pi }{2}\right[x\left]\right)+\left[x\right]+[x+\frac{1}{3}]$ $+[x+\frac{2}{3}]-3x$ is
$$\left[\begin{array}{c}where[.]\to \text{the greatest integer function}\end{array}\right]$$

• A. 8
• B. 2
• C. $\frac{1}{3}$
• D. Not periodic

Answer 1

The correct option is A 8

$Sin\left(\frac{\pi }{4}\right[x\left]\right)\to$ Period = 8
$tan\left(\frac{\pi }{2}\right[x\left]\right)\to$ period 2
$\left[x\right]+[x+\frac{1}{3}]+[x+\frac{2}{3}]=\left[3x\right]$
So $\left[3x\right]-3x=-\left\{3x\right\}\to$ period $=\frac{1}{3}$
where {.} denotes the fraction part function
So, the LCM of {8,2, $\frac{1}{3}$} is 8
So, the fundamental period of f is 8