How to Find the Critical Numbers for a Function

By Mark Ryan

All local extrema occur at critical points of a function — that’s where the derivative is zero or undefined (but don’t forget that critical points aren’t always local extrema). So, the first step in finding a function’s local extrema is to find its critical numbers (the x-values of the critical points).

Here’s an example: Find the critical numbers of f (x) = 3x5 – 20x3, as shown in the figure.

Here’s what you do:

  1. Find the first derivative of f using the power rule.


  2. Set the derivative equal to zero and solve for x.


These three x-values are critical numbers of f. Additional critical numbers could exist if the first derivative were undefined at some x-values, but because the derivative, 15x4– 60x2, is defined for all input values, the above solution set, 0, –2, and 2, is the complete list of critical numbers. Because the derivative of f equals zero at these three critical numbers, the curve has horizontal tangents at these numbers. In the figure, you can see the little horizontal tangent lines drawn where x = –2 and x = 2. The third horizontal tangent line where x = 0 is the x-axis.

A curve has a horizontal tangent line wherever its derivative is zero, namely, at its stationary points. A curve will have horizontal tangent lines at all of its local mins and maxes (except for sharp corners) and at all of its horizontal inflection points.

Now that you’ve got the list of critical numbers, you need to determine whether peaks or valleys or inflection points occur at those x-values. You can do this with either the first derivative test or the second derivative test. You may be wondering why you have to test the critical numbers when you can see where the peaks and valleys are by just looking at the graph in the figure — which you can, of course, reproduce on your graphing calculator. Good point. Okay, so this problem — not to mention countless other problems you’ve done in math courses — is somewhat contrived and impractical. So what else is new?