Question

At the point x = 2, the function

$f\left(x\right)=\{\begin{array}{cc}{x}^3-8& 2<x<\infty \\ x-2& -\infty <x\le 2\end{array}$ is

• A. continuous and differentiable
• B. continuous and not differentiable
• C. discontinuous and differentiable
• D. discontinuous and not differentiable

Answer 1

The correct option is B continuous and not differentiable

$\underset{x\to 2^{-}}{lim}f\left(x\right)=\underset{x\to 2^{-}}{lim}(x-2)=0$

$\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{x\to 2^{+}}{lim}(x^3-8)=0$

$\text{Also}f\left(2\right)=0$

$\text{Thus}\underset{x\to 2^{-}}{lim}f\left(x\right)=\underset{x\to 2^{+}}{lim}f\left(x\right)=f\left(2\right)$

$\therefore f\text{is continuous at}x=2$

$\text{and}L{f}^{\prime }\left(2\right)=1\text{and}R{f}^{\prime }\left(2\right)=12$

$\therefore f\text{is not differentiable at}x=2.$