Answer
a) $f(x)$ is differentiable at all points in $[-3,2]$.
b) There are no domain points in $[-3,2]$ where $y=f(x)$ is continuous but not differentiable.
c) There are no domain points where $y=f(x)$ is neither continuous nor differentiable.
Work Step by Step
*Some things to remember about differentiability:
- If $f(x)$ is differentiable at $x=c$, then $f(x)$ is continuous at $x=c$. (Theorem 1)
- $f(x)$ is not differentiable at $x=c$ if the secant lines passing $x=c$ fail to take up a limiting position or can only take up a vertical tangent. In other words, we can look at differentiability as the ability to draw a tangent line at a point, or the smoothness of the graph.
a) In this exercise, we have a continuous line on the closed interval $[-3,2]$. The tangent line of a line at any point is itself, meaning that at any point in $[-3,2]$, we can draw a tangent line for the graph of $y=f(x)$.
Therefore, $y=f(x)$ is differentiable at all points in $[-3,2]$.
b) There are no domain points in $[-3,2]$ where $y=f(x)$ is continuous but not differentiable.
c) There are no domain points where $y=f(x)$ is neither continuous nor differentiable.
