4. a) where is f(x) continuous but not differentiable? b) where else is f(x) not differentiable? y = f(x)
4. a) where is f(x) continuous but not differentiable? b) where else is f(x) not differentiable? y = f(x)
Answer
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 $x = 2$