Question

1. the figure below shows the graph of a function y = f(x) over the closed interval -1,2.

a) at what domain points does the function appear to be differentiable?
b) at what domain points does the function appear to be continuous but not differentiable?
c) at what domain points does the function appear to be neither continuous nor differentiable?
d) is y = f(x) a differentiable function? why or why not?

1. a function does not have a derivative or is not differentiable at a point if there is a __, , or __.

Explanation:

Step1: Recall differentiability and continuity concepts

A function is differentiable at a point if the graph has a smooth curve at that point. It is continuous if there are no breaks, jumps or holes.

Step2: Analyze differentiability for part a

The function is differentiable where the graph is smooth. For the given function $y = f(x)$ on $[-1,2]$, it is smooth on the open - intervals $(-1,0)$ and $(0,2)$. So the domain points where it is differentiable are $x\in(-1,0)\cup(0,2)$.

Step3: Analyze continuity and non - differentiability for part b

A function is continuous but not differentiable at a point where there is a corner or a cusp. For the given function, at $x = 0$ the function is continuous (no break in the graph) but has a corner, so it is continuous but not differentiable at $x=0$.

Step4: Analyze non - continuity and non - differentiability for part c

There are no points in the domain $[-1,2]$ where the function is neither continuous nor differentiable as the graph has no jumps, holes or asymptotes in the given domain.

Step5: Answer part d

The function $y = f(x)$ is not a differentiable function because it is not differentiable at $x = 0$ (it has a corner at $x = 0$).

Answer:

a) $x\in(-1,0)\cup(0,2)$
b) $x = 0$
c) None
d) No, because it has a corner at $x = 0$.