Chapter 3.2, Problem 9E

(a)

To determine

The domain points of the given function appear to be differentiable.

Answer to Problem 9E

The required domain points of the given function appears to be differentiable is $\left[-1,0\right)\cup \left(0,2\right]$ .

Explanation of Solution

Given information:

The given function is: $y=f\left(x\right)$ .

And, the domain points is $D:-1\le x\le 2$

  

The given function is: $y=f\left(x\right)$ .and the domain points is $D:-1\le x\le 2$ .

Since, from the figure it is seen that the function $y=f\left(x\right)$ has discontinuity point at $x=0$ .

Also, from the figure it is seen that the function $y=f\left(x\right)$ has vertical tangents at $x=0$ .

Thus, the function $y=f\left(x\right)$ is differentiable at all points in its domain except $x=0$ .

Hence, the required domain of the function appears to be differentiable is $\left[-1,0\right)\cup \left(0,2\right]$ .

(b)

To determine

The domain points of the given function appear to be continuous but not differentiable.

Answer to Problem 9E

The required domain of the given function appears to be continuous but not differentiable is $x=0$ .

Explanation of Solution

Given information:

The given function is: $y=f\left(x\right)$ .

And, the domain points is $D:-1\le x\le 2$

  

The given function is $y=f\left(x\right)$ .and the domain points is $D:-1\le x\le 2$ .

From the results (a) it is seen that the function $y=f\left(x\right)$ is differentiable at all points in its domain except $x=0$ .

Since, the given function has discontinuities at $x=0$ ..

Also, the given function $y=f\left(x\right)$ has vertical tangents at $x=0$ .

Thus, the given function $y=f\left(x\right)$ is continuous but not differentiable at $x=0$ .

Hence, the required domain of the function whether the function appears to be continuous but not differentiable is $x=0$ .

(c)

To determine

The domain of the given function appear neither continuous nor differentiable.

Answer to Problem 9E

The required domain of the given function appears to be neither continuous nor differentiable is none.

Explanation of Solution

Given information:

The given function is: $y=f\left(x\right)$ .

And, the domain points is $D:-1\le x\le 2$

  

The given function is $y=f\left(x\right)$ .and the domain points is $D:-1\le x\le 2$ .

From the graph it is seen that the given function $y=f\left(x\right)$ has no points of discontinuity.

Thus, there are no points where the function is neither continuous nor differentiable.

Hence, required domain of the given function appears neither continuous nor differentiable is none.