CPM Homework Help : CALC3RD Problem 7-68

No calculator! Determine where the following functions are not continuous and/or non-differentiable.

Compute without a calculator

$f(x) =\left\{ \begin{array} { l l } { \frac { 1 } { 2 } x ^ { 2 } + \frac { 3 } { 2 } } & { \text { for } x \leq - 1 } \\ { | 3 - x | } & { \text { for } x > - 1 } \end{array} \right.$
Hint (a):
Check continuity: Does $\lim\limits_{x\rightarrow -1^{-}}f(x)=\lim\limits_{x\rightarrow -1^{+}}f(x)=f(-1)?$
Hint (a):
Check differentiability: Does $\lim\limits_{x\rightarrow -1^{-}}f'(x)=\lim\limits_{x\rightarrow -1^{+}}f'(x)=f'(-1)?$
Also, recall that differentiability implies continuity.
$g(m) =\left\{ \begin{array} { l l l } { \frac { 2 } { m - 3 } } & { \text { for } } & { m < 2 } \\ { - \sqrt { 6 - m } } & { \text { for } } & { 2 \leq m < 5 } \\ { \frac { 1 } { 2 } m - \frac { 7 } { 2 } } & { \text { for } } & { m \geq 5 } \end{array} \right.$
Hint (b):
Refer to the hints in part (a).