Answer to Question #314789 in Real Analysis for Jyo

Question

Determine whether the following functions are differentiable

i) 𝑓 (𝑥 )= |𝑥|;

ii) 𝑔(𝑥) = |𝑥| +| 𝑥 + 1 |

iii) h(x) = x^1/3

Answer 1

$i:\\\underset{x\rightarrow 0+}{\lim}\frac{f\left( x \right) -f\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{x}{x}=1\\\underset{x\rightarrow 0-}{\lim}\frac{f\left( x \right) -f\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{-x}{x}=-1\\Not\,\,differentiable\\ii:\\\underset{x\rightarrow 0+}{\lim}\frac{g\left( x \right) -g\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{x+1-1}{x}=1\\\underset{x\rightarrow 0-}{\lim}\frac{g\left( x \right) -g\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{-x+1-1}{x}=-1\\Not\,\,differentiable\\iii:\\\underset{x\rightarrow 0}{\lim}\frac{h\left( x \right) -h\left( 0 \right)}{x-0}=\underset{x\rightarrow 0+}{\lim}\frac{x^{1/3}}{x}=\infty \\Not\,\,differentiable$