#### Please Solve R.D.Sharma class 12 Chapter 9 Differentiability Exercise 9.2 Question 2 Maths Textbook Solution.

Answer: f(x) is not differentiable at x = 0.

Hint: For differentiability, LHD = RHD where LHD is left hand derivative and RHD is right hand derivative. Also, LHD and RHD should exist in given limit.

Given: $f\left ( x \right )=x^{\frac{1}{3}}$

Solution:

As We Know

\begin{aligned} &f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &\text { LHD at } x=0: \lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0} \\ &=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{0-h-0} \\ &=\lim _{h \rightarrow 0} \frac{(-h)^{\frac{1}{3}}}{-h} \\ &=\lim _{h \rightarrow 0} \frac{(-1)^{\frac{1}{3}} h^{\frac{1}{3}}}{(-1) h} \\ &=\lim _{h \rightarrow 0}(-1)^{\frac{-2}{3}} h^{\frac{-2}{3}} \end{aligned}

$\Rightarrow Not \: \: defined$

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Answer: f(x) is not differentiable at x = 0.

Hint : For differentiability, LHD = RHD where LHD is left hand derivative and RHD is right hand derivative. Also, LHD and RHD should exist in given limit.

Given: $f\left ( x \right )=x^{\frac{1}{3}}$

Solution:

As We Know ,

\begin{aligned} &f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &\text { LHD at } \mathrm{x}=0: \lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0} \\ &=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{0-h-0} \\ &=\lim _{h \rightarrow 0} \frac{(-h)^{\frac{1}{3}}}{-h} \end{aligned}

\begin{aligned} &=\lim _{h \rightarrow 0} \frac{(-1)^{\frac{1}{3}} h^{\frac{1}{3}}}{(-1) h} \\ &=\lim _{h \rightarrow 0}(-1)^{\frac{-2}{3}} h^{\frac{-2}{3}} \end{aligned}                                                                                                                    $\left \{ \frac{a^{m}}{a^{n}}\Rightarrow a^{m-n} \right \}$

$\Rightarrow Not \: Defined$

\begin{aligned} &\mathrm{RHD} \text { at } \mathrm{x}=0: \lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0} \\ &=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{0+h-0} \\ &=\lim _{h \rightarrow 0} \frac{h^{\frac{1}{3}}}{h} \\ &=\lim _{h \rightarrow 0} h^{\frac{2}{3}} \end{aligned}                                                                                    $\left \{ \frac{a^{m}}{a^{n}}\Rightarrow a^{m-n} \right \}$

$\Rightarrow Not \: Defined$

LHD and RHD does not exist at x = 0.

Hence, f(x) is not differentiable at x = 0.