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  4. The exhaustive set of values of x for which f(x)=√[5]x2|x|3-√[3]x2|x|-1 is not differentiable is

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Q. The exhaustive set of values of $x$ for which $f(x)=\sqrt[5]{x^{2}|x|^{3}}-\sqrt[3]{x^{2}|x|}-1$ is not differentiable is

Continuity and Differentiability

Solution:

$f(x)=\sqrt[5]{x}^{2}|x|^{3}-\sqrt[3]{x^{2}|x|-1}$
$\Rightarrow f(x) = \begin{cases} \left(x^{5}\right)^{\frac{1}{5}}-\left(x^{3}\right)^{\frac{1}{3}}-1, & \text{ $x \geq \,0$ } \\[2ex] \left(-x^{5}\right)^{\frac{1}{5}}-\left(-x^{3}\right)^{\frac{1}{3}}-1, & \text{$x<\,0$ } \end{cases}$
$ = \begin{cases} x-x-1, & \text{if $x \geq \,0$} \\[2ex] -x+x-1, & \text{if $x<\,0$ } \end{cases}$
$=-1$ for all x
Thus, f(x) is differentiable everywhere