Chapter 1.3, Problem 44E

To determine

To calculate: To graph the function and to approximate any relative minimum or maximum values of the function

Answer to Problem 44E

The function has a relative minimum at the point $\left(0,0\right)$ and relative maximum at $\left(2,4\right)$

Explanation of Solution

Given: Function is $f\left(x\right)=-x^3+3x^2$

Calculation:

Function is given as follows:

  $f\left(x\right)=-x^3+3x^2$

When $x=-1$

  $\begin{array}{l}f\left(-1\right)=-{\left(-1\right)}^3+3{\left(-1\right)}^2\\ f\left(-1\right)=-1+3\\ f\left(-1\right)=2\end{array}$

When $x=0$

  $\begin{array}{l}f\left(0\right)=-{\left(0\right)}^3+3{\left(0\right)}^2\\ f\left(0\right)=30\end{array}$

When $x=2$

  $\begin{array}{l}f\left(2\right)=-{\left(2\right)}^3+3{\left(2\right)}^2\\ f\left(2\right)=-8+12\\ f\left(2\right)=4\end{array}$

On plotting the above points on graph:

  

From the above graph, function is increasing in the interval $\left(0,2\right)$ and the function is decreasing in the interval $\left(-\infty ,0\right)$ and $\left(2,\infty \right)$

Thus, function has a relative minimum at the point $\left(0,0\right)$ and relative maximum at $\left(-1,2\right)$

Conclusion:

Hence, graph of function is sketched and the function has a relative minimum at the point $\left(0,0\right)$ and relative maximum at $\left(2,4\right)$