Limit Continuity And Differentiability Ques 15
- Let $f(x)=x|x|$ The set of points, where $f(x)$ is twice differentiable, is ……. .
(1992, 2M)
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Answer:
Correct Answer: 15.$(x \in R-\{0\})$
Solution: Given, $f(x)=x|x|$
$\Rightarrow \quad f(x)= \begin{cases}x^2, & \text { if } x \geq 0 \\ -x^2, & \text { if } x<0\end{cases}$
$f(x)$ is not differentiable at $x=0$ but all $R-\{0\}$.
Therefore, $\quad f^{\prime}(x)= \begin{cases}2 x, & x>0 \\ -2 x, & x<0\end{cases}$
$\Rightarrow \quad f^{\prime \prime}(x)= \begin{cases}2, & x>0 \\ -2, & x<0\end{cases}$
Therefore, $ \quad f(x)$ is twice differentiable for all $x \in R-\{0\}$.
BETA
