Curve Sketching and Extrema

In this exercise, we are going to analyze and sketch the curve of the function f of x equals x cubed minus three x. We will focus on finding the critical points to identify the extreme values.

To find the stationary points or extreme values, we first need to find the derivative of the function.

Using the power rule, the derivative of x cubed is three x squared, and the derivative of minus three x is minus three.

Extrema occur where the first derivative is equal to zero. So, let's set our derivative to zero and solve for x.

We can factor out a three, which gives us three times the quantity x squared minus one equals zero.

This implies x squared equals one, meaning x can be positive one or negative one. These are our critical x-values.