Chapter 10, Problem 27RE

(a)

To determine

The values of x for which the graph of $f\left(x\right)$ is increasing, the graph of $f\left(x\right)$ is decreasing and where it has relative extrema from the graph of the first derivative function $f\text{'}\left(x\right)=x^2-4x-5$.

(b)

To determine

The values of x for which the $f^{″}\left(x\right)>0,f^{″}\left(x\right)<0\text{ and }{f}^{″}\left(x\right)=0$ from the graph of the first derivative function $f\text{'}\left(x\right)=x^2-4x-5$.

(c)

To determine

To prove: The function $f\left(x\right)=\frac{x^3}{3}+2x^2-5x$ has its derivative equal to $f^{\prime }\left(x\right)=x^2+4x^2-5$ and also sketch the graph of the function $f\left(x\right)=\frac{x^3}{3}+2x^2-5x$ and check the conclusion.

(d)

To determine

Whether the conclusions made in part (b) are correct by graphing the second derivative of the provided function $f\left(x\right)=\frac{x^3}{3}+2x^2-5x$.