Definitions
A relative maximum is a point at which a function attains its highest value relative to the points nearby. Visually, this occurs at any 'peak' on the function's graph.
Similarly, a relative minimum is a point at which a function attains its lowest value relative to points nearby.
Together, relative maxima (plural of maximum) and minima (plural of minimum) are known as the relative extrema of the function. (All those weird words are actually in Latin, but don't worry, we won't be conjugating any verbs today!) A given function could have many extrema. Any peak or valley in the graph counts.
The graphs of sin(x) and cos(x) both have infinitely many relative maxima and minima!
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As you can probably imagine, these kinds of points are of interest in a variety of settings. For example, if your company has a function that models its profit based on how many units it produces, you would certainly be interested in that function's highest points!
Finding Relative Extrema
The key to finding these points is this fact: if f(x) is a nice smooth function (no breaks or corners), then at any maximum or minimum (extremum), when the graph is turning around, there must be a horizontal tangent at that point. And this means that the derivative value will be zero at that point, because a horizontal tangent has slope equal to 0.
There are relative extrema at each point where the graph has a horizontal tangent.
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Example 1
Let's do an example: find the relative maximum and minimum of the function
This is the function whose graph is pictured near the top of this article. The 'maximum' or 'minimum' feature under the CALC menu on your trusty Texas Instruments calculator would tell you that the maximum and minimum occur around x=-0.577 and x=0.577 respectively. However, our skills in finding derivatives will help us find exactly where these points lie. There is an easy 3-step method.
- Step 1: Find the derivative of f.
Remembering the power rule for derivatives, we find that:
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- Step 2: Set the derivative equal to zero and solve for x (this part is just algebra).
- Step 3: If the problem asks for the points at which the extrema occur, then we have to find the y-values at each of these points by plugging our x-values into f(x).
And now we know where the relative extrema occur on this graph! Let's do another.
Example 2
Find the relative minimum of f(x)=x ln(x).
- Step 1: We're going to need the product rule for derivatives to find f'(x), and don't forget the derivative of ln(x) is 1/x.
- Step 2: Now set f'(x) equal to zero and solve for x.
Note, 1/e is equal to .3678794412...
- Step 3: Find the y-value.
Thus, the relative minimum occurs at (approximately) the point (0.368, -0.368).
Lesson Summary
Relative extrema is a fancy term for the maximum and minimum points of a graph, relative to nearby points. To find the exact values (coordinates) for these points, use the three step process:
- Step 1: Find the derivative of f.
- Step 2: Set the derivative equal to 0 and solve for x.
- Step 3: Find the y-value(s) by plugging in each x-value into the original function f.
