Local maximum

Definition

A local maximum is a point on the graph of a function where the function value is greater than or equal to the values of the function at nearby points. It represents a peak in a specific region of the graph.

5 Must Know Facts For Your Next Test

1. A local maximum occurs where the first derivative of the function changes from positive to negative.
2. The second derivative test can help confirm if a critical point is a local maximum: if $f''(x) < 0$ at that point, it is a local maximum.
3. Local maxima are not necessarily the highest points on the entire graph; they are only higher than neighboring points.
4. In polynomial functions, local maxima occur at critical points where $f'(x) = 0$ or $f'(x)$ does not exist.
5. $f(x)$ has a local maximum at $x = c$ if $f(c) \geq f(x)$ for all $x$ near $c$.

Review Questions

• What is indicated by a change in sign from positive to negative in the first derivative?
• How can you use the second derivative test to determine if a point is a local maximum?
• Explain why a local maximum might not be the highest point on an entire graph.

Related terms

Critical Point: A point on the graph where the first derivative is zero or undefined, which could be a location for local maxima, minima, or saddle points.

First Derivative Test: A method used to determine if a critical point is a local minimum or maximum by analyzing changes in sign of the first derivative around that point.

Second Derivative Test: A test using the second derivative to determine whether a critical point is a local minimum (if $f''(x) > 0$) or maximum (if $f''(x) < 0$).