*Shaun Ault*Show bio

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

Instructor:
*Shaun Ault*
Show bio

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

Finding the points at which a function attains its highest or lowest value has many practical applications. In this lesson, you will learn how to use basic calculus to find the relative minimums and maximums of any function. Only basic knowledge of finding derivatives and solving equations using algebra will be necessary to understand this topic.

Where is the best pizza? Depends on who you ask, right? When I think of the best pizza, I'm really just thinking of the best in my city or neighborhood. Let's say it's 'Bob's Pizza.' So Bob's might have the best pizza out of all pizza places *near* me, but not the best pizza on the planet. **Relative** to my location, Bob's pizza is a **maximum**.

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.

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!

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.

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:

- 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.

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).

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*.

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