Graph the function. Label all intercepts, critical points, and inflection points.

*f* (*x*) = *x*^{3} - 5*x*

Answer

*f* (*x*) = *x*^{3} - 5*x*

• Find dots.

To find the roots, we factor the function:

*f* (*x*) = *x*(*x*^{2} - 5)

The roots are *x* = 0 and . The root *x* = 0 is also the *y*-intercept.

The derivative of *f* is

*f*'(*x*) = 3*x*^{2} - 5.

This is zero when

so these are our critical points. For the sake of graphing, The corresponding *y* values are

The second derivative is

*f*"(*x*) = 6*x*.

This is 0 when *x* = 0, so that's our inflection point.

We graph the dots, labeling their type. So long as it's written down somewhere what the coordinates of the critical points are, don't worry too much about trying to fit that label on the graph.

• Find shapes.

Here are the numberlines:

*f* (*x*) = x(*x*^{2} - 5) is negative when *x* is negative and *x*^{2}-5 is positive, which occurs when . *f* is also negative when *x* is positive and (*x*^{2} - 5) is negative, which occurs when . *f* is positive everywhere else.

*f*'(*x*) = 3*x*^{2} - 5 is negative when and positive everywhere else.

*f*"(*x*) = 6*x* is negative when *x* is negative and positive when *x* is positive.

Now we can find the shapes:

• Play connect-the-dots.

Here's the graph: