Relative Extrema#
Definition
The relative extrema of a function \(f\):
A function \(f\) has a relative maximum at \(x=c\) if there exists an open interval \((a,b)\) containing \(c\) such that \(f(c) \geq f(x)\) for all \(x\) in \((a,b)\).
A function \(f\) has a relative minimum at \(x=c\) if there exists an open interval \((a,b)\) containing \(c\) such that \(f(c) \leq f(x)\) for all \(x\) in \((a,b)\).
Example 1#
The relative extrema are highlighted on the following graph. Observe how the relative extrema appear at points on the curve where the increasing/decreasing behavior of the function changes.
Long Text Description
There is a horizontal x axis. There is a vertical y axis. The graph of a function is plotted on these axes. Moving from left to right, the function goes up towards a rounded corner, which is marked in red and labeled a relative maximum, then goes downward towards a sharp corner, which is marked in green and labeled a relative minimum, then goes up again towards another sharp corner, which is marked in red and labeled a relative maximum, then goes down again towards a rounded corner, which is marked in green and labeled a relative minimum, and then goes upward.
In other words, relative extrema appear at points on the graph of the function where the derivative changes sign.
How To Find Relative Extrema#
The relative extrema of a function appear where \(f'(x)\) changes from positive to negative or from negative to positive. Since \(f'(x)\) changes sign when there is a relative extrema, it must be the case that either \(f'(x)=0\) or \(f'(x)\) does not exist at the relative extrema.