### Home > CALC > Chapter 4 > Lesson 4.1.3 > Problem 4-38

4-38.

On your paper, sketch a graph of

*f*(*x*) =*x*^{3}+ 3*x*^{2}− 45*x*+ 8. Homework Help ✎Find the slope of the line tangent to the curve at

*x*= −2.Find the point on the curve where the slope is the smallest (steepest negative slope). What is the name of this point?

Recall that the slope of the tangent line at *x* = *a* is also known as *f* '(*a*).

You are looking for the location where *f* '(*x*) is at its lowest (or at a minimum). To find this location, you will need to find the location (*x*-value) of the lowest point (the vertex) on the graph of *f* '(*x*). Note: *f* '(*x*) is a parabola, so an Algebra I student could complete this part of the task.

Use the *x*-value of the vertex of *f* '(*x*), to evaluate the coordinate point on *f*(*x*) where the slope is the steepest.