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

MCQ

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

✓

Answer

Correct option: A.

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

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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