Chapter 1.3, Problem 43E

To determine

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

Answer to Problem 43E

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

Explanation of Solution

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

Calculation:

Function is given as follows:

  $f\left(x\right)=x^3-3x$

When $x=-1$

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

When $x=0$

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

When $x=1$

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

On plotting the above points on graph:

  

From the above graph, function is increasing in the interval $\left(-\infty ,-1\right)$ and $\left(1,\infty \right)$ and the function is decreasing in the interval $\left(-1,1\right)$

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