Consider f(x) = l x^2 |x| , ,x 0 , , , , , , ,0, ,x = 0 . - Vidyadip

MCQ

Consider $f(x) = \left\{ \begin{array}{l}\frac{{{x^2}}}{{|x|}},\,x \ne 0\\\,\,\,\,\,\,\,0,\,x = 0\end{array} \right.$

A
$f(x)$ is discontinuous everywhere
B
$f(x)$ is continuous everywhere
C
$f'(x)$ exists in $( - 1,1)$
D
$f'(x) $ exists in $( - 2,2)$

✓

Answer

We have $f(x) = \left\{ {\begin{array}{*{20}{r}}{\frac{{{x^2}}}{{|x|}},}&{x \ne 0}\\{0,}&{x = 0}\end{array} = \left\{ {\begin{array}{*{20}{r}}{\frac{{{x^2}}}{x} = x,}&{x > 0}\\{0,}&{x = 0}\\{\frac{{{x^2}}}{{ - x}} = - x,}&{x < 0}\end{array}} \right.} \right.$
We have $\mathop {\lim }\limits_{x \to 0 - } f(x) = \mathop {\lim }\limits_{x \to 0} \, - x = 0,\mathop {\lim }\limits_{x \to 0 + } f(x) = \mathop {\lim }\limits_{x \to 0} x = 0$
and $f(0) = 0.$
So $f(x)$ is continuous at $x = 0.$
Also $f(x)$ is continuous for all other values of $x.$
Hence, $f(x)$ is continuous everywhere.
Clearly, $Lf'(0) = - 1$ and $Rf'(0) = 1.$
Therefore $f(x)$ is not differentiable at $x = 0.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Which of the following are true ?

If $3$ distinct real number $a$,$b$,$c$ satisfy $a^2(a + p) = b^2 (b + p) = c^2 (c + p)$ where $p \in R$, then value of $bc + ca + ab$ is

Let $S=\left\{\frac{a^2+b^2+c^2}{a b+b c+c a}: a, b, c \in R, a b+b c+c a \neq 0\right\}$ where $R$ is the set of real numbers. Then, $S$ equals

The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^{2}}}{1-2 x^{2}}\right)$ at $x=\frac{1}{2}$ is

${\tan ^{ - 1}}\frac{3}{4} + {\tan ^{ - 1}}\frac{3}{5} - {\tan ^{ - 1}}\frac{8}{{19}} = $

Let the vectors $(2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k}$ $(1+\mathrm{b}) \hat{i}+2 \mathrm{b} \hat{j}-\mathrm{b} \hat{k}$ and $(2+\mathrm{b}) \hat{i}+2 \mathrm{b} \hat{j}+(1-\mathrm{b}) \hat{k}$ $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{R}$ be co$-$planar. Then which of the following is true$?$

$\int_{}^{} {\sqrt {1 + \sin x} \;dx = } $

If $5x + 9 = 0$ is the directrix of the hyperbola $16x^2 -9y^2 = 144,$ then its corresponding focus is

The number of distinct real roots of the equation $|\mathrm{x}+1||\mathrm{x}+3|-4|\mathrm{x}+2|+5=0$, is ...........

The value of $f$ at $x = 0$ so that the function $f(x) = \frac{{{2^x} - {2^{ - x}}}}{x},x \ne 0$, is continuous at $x = 0$, is