MCQ
The function $f(x)=|x|$ is
- Acontinuous and differentiable everywhere.
- Bcontinuous and differentiable nowhere.
- ✓continuous everywhere, but differentiable everywhere except at $x=0$.
- Dcontinuous everywhere, but differentiable nowhere.
✓
Answer
Correct option: C.
continuous everywhere, but differentiable everywhere except at $x=0$.
$f(x)=|x|$ = $\left\{\begin{aligned} x, & x>0 \\ -x, & x<0\end{aligned}\right.$
The function $f(x)$ is continuous everywhere but not differentiable at $x=0$ as at $x=0$
$L f^{\prime}(0)=\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}$
$=\lim _{x \rightarrow 0} \frac{-x-0}{x}=-1$
$R f^{\prime}(0)=\lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0}$
$=\lim _{x \rightarrow 0} \frac{x-0}{x}=1$
$\therefore L f^{\prime}(0) \neq R f^{\prime}(0),$
so $f(x)$ is not differentiable at $x=0$.
The function $f(x)$ is continuous everywhere but not differentiable at $x=0$ as at $x=0$
$L f^{\prime}(0)=\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}$
$=\lim _{x \rightarrow 0} \frac{-x-0}{x}=-1$
$R f^{\prime}(0)=\lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0}$
$=\lim _{x \rightarrow 0} \frac{x-0}{x}=1$
$\therefore L f^{\prime}(0) \neq R f^{\prime}(0),$
so $f(x)$ is not differentiable at $x=0$.
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