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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
Choose the correct answers from the given four options : Let $\text{f(x)}=|\sin\text{x}|.$ Then :
  • A
    $f$ is everywhere differentiable.
  • ✓
    $f$ is everywhere continuous but not differentiable at $\text{x}=\text{n}\pi,\text{n}\in\text{Z}.$
  • C
    $f$ is everywhere continuous but not differentiable at $\text{x}=(2\text{n}+1)\frac{\pi}{2},\text{n}\in\text{Z}.$
  • D
    None of these.
✓

Answer

Correct option: B.
$f$ is everywhere continuous but not differentiable at $\text{x}=\text{n}\pi,\text{n}\in\text{Z}.$
Let $\text{u(x)}=\sin\text{x}$ and $\text{v(x)}=|\text{x}|$
$\therefore\ \text{f(x)}=\text{vou(x)}=\text{v}[\text{u(x)}]$
Since, $u(x)$ and $v(x)$ both are continuous functions.
Hence, $f(x) = \text{vou(x)}$ is also a continuous function but $v(x)$ is not differentiable at $x = 0.$
So, $f(x)$ is not differentiable where $\sin\text{x}=0$
$\Rightarrow\ \text{x}=\text{n}\pi,\text{n}\in\text{Z}$
Hence, $f(x)$ is continuous everywhere but not differentiable at $\text{x}=\text{n}\pi,\text{n}\in\text{Z}.$

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