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Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferentiability1 Mark
Question
The function $\text{f(x)}=\frac{\sin(\text{x}|\text{x}-\pi|)}{4+|\text{x}|^2},$ where[.] denotes the greatest integer function, is:
  1. Continuous as well as differentiable for all $\text{x}\in\text{R}$
  2. Continuous for all x but differentiable at some x
  3. Differentiable for all x but not continuous at some x
  4. None of these.
✓

Answer

  1. 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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