Chapter 10, Problem 28RE

(a)

To determine

The region where the graph of $f\left(x\right)$ is increasing and decreasing and the location relative extrema in the graph of the function $f^{\prime }\left(x\right)=6x^2-x^3$.

(b)

To determine

To calculate: The region where $f^{″}\left(x\right)>0$, $f^{″}\left(x\right)<0$ and $f^{″}\left(x\right)=0$ in the graph of the function $f^{\prime }\left(x\right)=6x^2-x^3$.

(c)

To determine

To prove: The derivative $f^{\prime }\left(x\right)$ is the derivative of $f\left(x\right)$ and also check the conclusion from part (a) by draw the graph of $f\left(x\right)$.

(d)

To determine

To calculate: The function $f^{″}\left(x\right)$ for the function $f\left(x\right)=2x^3-\frac{x^4}{4}$ and check the conclusion from part (b) by draw the graph of $f^{″}\left(x\right)$.