$I f f (x) = | x |^{3}, s h o w t h a t f^{''} (x) e x i s t s f o r a l l x a n d f i n d i t .$

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Solution

$H e r e, f (x) = | x |^{3} W h e n x \leq 0, f (x) = | x |^{3} = x^{3} D i f f e r e n t i a t i n g b o t h s i d e s w . r . t . x, w e h a v e \frac{d}{d x} [f (x)] = \frac{d}{d x} (x^{3}) \Rightarrow f^{'} (x) = 3 x^{2} A g a i n d i f f e r e n t i a t i n g b o t h s i d e s w . r . t . x, w e h a v e \frac{d}{d x} [f^{'} (x)] = \frac{d}{d x} (3 x^{2}) \Rightarrow f^{''} (x) = 6 x W h e n x < 0, f (x) = | x |^{3} = - x^{3} D i f f e r e n t i a t i n g b o t h s i d e s w . r . t . x, w e h a v e \frac{d}{d x} [f (x)] = \frac{d}{d x} (- x^{3}) \Rightarrow f^{'} (x) = - 3 x^{2} A g a i n d i f f e r e n t i a t i n g b o t h s i d e s w . r . t . x, w e g e t \frac{d}{d x} [f^{'} (x)] = \frac{d}{d x} (- 3 x^{2}) - 6 x \frac{d}{d x} f^{''} (x) = - 6 x . H e n c e, f^{'} (x) = {\begin{matrix} 6 x, x \geq 0 - 6 x, x < 0 \end{matrix}$

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