Local Maximum and Minimum Classification

Local maximum and minimum classification

These exercises ask you to decide whether a critical point is a local maximum, a local minimum, or neither. The key idea is to study how the function behaves near the point.

1) Find the point to test

A local-extrema classification problem usually gives a point, or asks you to use derivative information at a candidate point. Make sure you know which point you are classifying.

2) Use the sign of the first derivative nearby

Check the sign of the derivative just to the left and right of the point:

• If the derivative changes from positive to negative, the function rises then falls, so the point is a local maximum.
• If the derivative changes from negative to positive, the function falls then rises, so the point is a local minimum.
• If the derivative does not change sign, the point is neither.

3) If second-derivative information is given

When a second derivative test is available:

• Positive second derivative suggests a local minimum.
• Negative second derivative suggests a local maximum.
• If the second derivative gives no clear conclusion, use another test.

4) Write the final classification clearly

State the result directly and simplify any expression if the answer includes one.

Check

A good check is to compare the function values or slope behavior on each side of the point. The answer should match the local shape of the graph.