Question

g(x)=0.5∣x+2∣

Answer 1

Given g(x)=0.5|x+2|

Now,

$$g(x)=\begin{cases}0.5(x+2), \text{if} \:x\ge -2\\-0.5(x+2),\text{if}\: x<-2\end{cases}$$

We have to check whether g(x) is differentiable at -2.

Clearly, left hand derivative of g(x) at -2 is $g'(-2^{-})=-0.5$ and right hand derivative is $g'(-2^{+})=0.5$ but $g'(-2^{-})\ne g'(-2^{+})$

Hence, g is not differentiable at -2