The function f(x)= (x|x- |) 4+|x|^2 , where[.] denotes the greate… - Vidyadip

Question

The function $\text{f(x)}=\frac{\sin(\text{x}|\text{x}-\pi|)}{4+|\text{x}|^2},$ where[.] denotes the greatest integer function, is:

Continuous as well as differentiable for all $\text{x}\in\text{R}$
Continuous for all x but differentiable at some x
Differentiable for all x but not continuous at some x
None of these.

✓

Answer

Continuous as well as differentiable for all $\text{x}\in\text{R}$

Solution:
Here,
$\text{f(x)}=\frac{\sin(\text{x}|\text{x}-\pi|)}{4+|\text{x}|^2}$
Since, we know that $\pi(\text{x}-\pi)=\text{n}\pi$ and $\sin\text{n}\pi=0.$
$\because4+\text{x}[\text{x}]^2\neq0$
$\therefore\text{f(x)}=0$ for all x
Thus, f(x) is a constant function and it is continuous and differentible everywhere.

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