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Please solve RD Sharma class 12 chapter 9 Differentiability exercise Very short answer type question  9 maths textbook solution

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Hint :  If f(x) is differentiable at x=0 then \lim _{x \rightarrow 0} \frac{f(x)-f(0)}{x-0}=f^{\prime}(0)  exists.


Explanation:  \mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc} x^{3} & x \geq 0 \\ -x^{3} & x<0 \end{array}\right.

        \begin{aligned} &\text { LHD of } \mathrm{x}=0\\ &\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}=\lim _{x \rightarrow 0} \frac{-x^{3}-0}{x}=0\\\\ &R H D \text { at } x=0\\\\ &\lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0}=\lim _{x \rightarrow 0} \frac{x^{3}-0}{x}=0 \end{aligned}

        \begin{aligned} &\text { As } R H D=L H D\\\\ &f(x) \text { is differentiable at } x=0\\\\ &f^{\prime}(0)=\lim _{x \rightarrow 0} \frac{f(x)-f(0)}{x-0}=0 \end{aligned}

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