How to Find Local Maximum and Minimum Points Using Derivatives

A local maximum is a point where a function value is greater than nearby values. A local minimum is smaller than nearby values. Derivatives identify candidates and help classify them.

Step 1: find critical numbers

Calculate f′(x). Critical numbers occur where f′(x)=0 or where the derivative is undefined while the original function is defined.

Step 2: use the first-derivative test

• If f′ changes from positive to negative, f has a local maximum.
• If f′ changes from negative to positive, f has a local minimum.
• If the sign does not change, the point is not a local extremum.

Worked example

Let f(x)=x^3-3x+1.

f′(x)=3x^2-3=3(x-1)(x+1)

The critical numbers are x=-1 and x=1. The derivative sign pattern is positive, negative, positive. Therefore:

• x=-1 gives a local maximum.
• x=1 gives a local minimum.

Evaluate the original function:

• f(-1)=3, so the local maximum point is (-1,3).
• f(1)=-1, so the local minimum point is (1,-1).

The second-derivative test

If f′(c)=0 and f″(c)>0, the graph is concave up and c is a local minimum. If f″(c)<0, the graph is concave down and c is a local maximum.

For the example, f″(x)=6x. Thus f″(-1)<0 confirms a maximum and f″(1)>0 confirms a minimum.

When the second-derivative test is inconclusive

If f″(c)=0 or does not exist, the test gives no conclusion. Return to the first-derivative sign test or use another argument. A stationary point can be an inflection point rather than an extremum.

Local versus absolute extrema

Local extrema compare nearby values. Absolute extrema compare all values on the domain or interval. On a closed interval, also evaluate the endpoints when searching for absolute maxima and minima.

Use the online analysis tools

The Graphing Calculator can display the curve and support point analysis. The Variation Table tool summarizes derivative signs and extrema in a structured table.

Common mistakes

• Reporting only the x-coordinate instead of the point.
• Assuming every solution of f′(x)=0 is an extremum.
• Forgetting derivative-undefined critical points.
• Using the second-derivative test when f″(c)=0.
• Confusing local and absolute extrema.

Create a variation table for the worked example →

To extend the analysis to concavity, continue with first- and second-derivative analysis.