A local maximum of a function is a point at which the function's value is higher than at any nearby points. It represents a peak within a specific interval.
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A local maximum occurs where the first derivative of the function changes from positive to negative.
If $f'(c) = 0$ and $f''(c) < 0$, then $f(c)$ is a local maximum.
Local maxima are found using critical points, where the first derivative is zero or undefined.
The second derivative test can help determine if a critical point is a local maximum.
Local maxima are not necessarily the highest points on the entire graph, only within their immediate vicinity.
Review Questions
How do you use the first derivative to identify potential local maxima?
What conditions must be met for a critical point to be classified as a local maximum using the second derivative test?
Why might a local maximum not be the global maximum of a function?
Related terms
Critical Point: A point on the graph where the first derivative is zero or undefined.
Global Maximum: The highest point over the entire domain of a function.
Second Derivative Test: A method used to determine if a critical point is a local minimum, local maximum, or neither by evaluating the second derivative at that point.