Question

Write the derivative of f (x) = |x|³ at x = 0.

Answer 1

Given:  

 

`f(x) = |x|^3 = {(x^3, xge0),(-x^3 , x<0):}`

(LHD at x = 0)

\[\lim_{x \to 0^-} \frac{f(x) - f(0)}{x - 0}\]
\[ = \lim_{h \to 0} \frac{f(0 - h) - f(0)}{x}\]
\[ = \lim_{h \to 0} \frac{h^3}{- h} \]
\[ = 0\]

(RHD at x = 0)

\[\lim_{x \to 0^+} \frac{f(x) - f(0)}{x - 0} \]
\[ = \lim_{x \to 0^+} \frac{f(0 + h) - f(0)}{x}\]
\[ = \lim_{h \to 0} \frac{h^3 - 0}{h} \]
\[ = 0\]

and 

\[f(0) = 0 .\]

Thus, (LHD at x=0) = (RHD at x = 0) = 

\[f(0)\]

Hence, 

\[\lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = f'(0) = 0\]