Question

1. a) where is f(x) continuous but not differentiable? b) where else is f(x) not differentiable? y = f(x)

Explanation:

Step1: Recall differentiability - continuity relation

A function is not differentiable at sharp - corners and vertical asymptotes. A function is continuous if there are no breaks in the graph.

Step2: Identify sharp - corners

At $x=- 2$, the function $y = f(x)$ has a sharp - corner. The left - hand and right - hand derivatives do not match at this point. The function is continuous at $x =-2$ (no break in the graph), but not differentiable.

Step3: Identify other non - differentiable points

At $x = 2$, the function has a vertical asymptote. The function is not continuous at $x = 2$ and thus not differentiable. Also, at $x=0$, the function has a cusp - like shape and is not differentiable.

Answer:

a) $x=-2$
b) $x = 0$ and $x = 2$