Chapter 2.3, Problem 5E

Each of the graphs of the functions in Exercises $1-8$ has one relative maximum and one relative minimum point. Find these points using the first-derivative test.

Use a variation chart as in Example 1.

$f(x)=\frac{1}{3}{x}^3-x^2+1$

Here is the chart:

Critical Values

$\begin{array}{cccccc}\text{Intervals}& x<1& x=1& 1<x<3& x=3& 3<x\\ f\text{'}\left(x\right)& +& 0& -& 0& +\\ f\left(x\right)& \text{Increasing}\text{on }\left(-\infty ,1\right)& \frac{7}{3}& \text{Decreasing on }\left(1,3\right)& 1& \text{Increasing on }\left(3,\infty \right)\\ & \nearrow & \begin{array}{l}\text{Local}\text{maximum}\\ \left(1,\frac{7}{3}\right)\end{array}& \searrow & \begin{array}{l}\text{Local}\text{minimum}\\ \left(3,1\right)\end{array}& \nearrow \end{array}$