Local maximum

Definition

A local maximum of a function is a point at which the function's value is higher than that of any nearby points. It is not necessarily the highest point on the entire graph, but rather within a specific interval.

5 Must Know Facts For Your Next Test

1. A local maximum occurs where the first derivative is zero and changes from positive to negative.
2. The second derivative test can confirm a local maximum if it is negative at that critical point.
3. A polynomial function of degree $n$ can have up to $n-1$ local maxima or minima.
4. Local maxima are useful for understanding the behavior and turning points of polynomial functions.
5. Examining intervals around critical points helps determine whether they are local maxima or minima.

Review Questions

• What condition must be met by the first derivative for a point to be considered as a potential local maximum?
• How does the second derivative test help in identifying a local maximum?
• If a cubic polynomial has three distinct real roots, how many local maxima can it have?

Related terms

Critical Point: A point on the graph where the first derivative is zero or undefined, indicating potential maxima, minima, or inflection points.

Local Minimum: A point where the function's value is lower than that of any nearby points within a specific interval.

Inflection Point: A point on the graph where the concavity changes from concave up to concave down, or vice versa.