Chapter 4.3, Problem 78E

To determine

(a)

To show:

The derivative of $f\left(x\right)=x^3\sin \left(\frac{1}{x}\right)$ is continuous at $x=0$ and that $x=0$ is a critical point of $f$.

To determine

(b)

To draw:

The graphs of function $f$ and $f^{\prime }$, also explain whether first derivative test is applied or not.

To determine

(c)

To show:

The function $f\left(0\right)$ is neither a local min nor a local max.