Study Guide

Extreme Points and How to Find Them

Extreme Points and How to Find Them

The maximum value of the function f (x) = -x2 – 1 is y = -1:

The maximum value of the function f (x) = cos x is y = 1:

Extreme points, also called extrema, are places where a function takes on an extreme value—that is, a value that is especially small or especially large in comparison to other nearby values of the function. Extrema look like the tops of hills and the bottoms of valleys. Time to go hiking.

There are two types of extreme points, minima (the valleys) and maxima (the hills).

Extreme points can be local or global, but we'll talk about this later.

We need to define minimum and maximum values without the on an interval bit.

A minimum value of a function is a y-value of the function that is as low, or lower, than other values of the function nearby. A minimum looks like a valley:

The plural of minimum is minima.

Sample Problem

The minimum value of the function f (x) = x2 + 1 is y = 1:

Sample Problem

The minimum value of the function f (x) = cos x is y = -1:

A function may have multiple minima.

Sample Problem

The function graphed below has two minima: y = 0 and y = 1.

A function may have infinitely many minima.

Sample Problem

The function graphed below has infinitely many minima:

A function may have no minima at all.

Sample Problem

The function f (x) = -x2 has no minima, because for every value of the function there are smaller values nearby:

Be Careful:There is a difference between a minimum of a function (a y-value) and where that minimum occurs (an x-value).

Sample Problem

The minimum value of the function f (x) = x2 + 1 is y = 1, and this minimum occurs at x = 0:

Sample Problem

The function f (x) = cos x has only one minimum value, y = -1. However, this minimum value occurs at infinitely many places, as it occurs at x = π + 2nπ for every integer n:

A function may have multiple maxima.

Sample Problem

The function graphed below has two maxima: y = 2 and y = 3.

A function may have infinitely many maxima.

Sample Problem

The function graphed below has infinitely many maxima:

A function may have no maxima at all.

Sample Problem

The function f (x) = x2 has no maxima, because for every value of the function there are larger values nearby:

Be Careful:There is a difference between a maximum of a function (a y-value) and where that maximum occurs (an x-value).

Sample Problem

The maximum value of the function f (x) = -x– 1 is y = -1, and this maximum occurs at x = 0:

Sample Problem

The function f (x) = cos x has only one maximum value, y = 1. However, this maximum value occurs at infinitely many places, as it occurs at x = 2πn for every integer n:

  • Finding & Classifying Extreme Points

    Stay classy, San Diego. If we have a function f that's defined on the whole real line, any extreme points must occur at critical points, as these are the only points at which we can have a peak or a valley:

    Any other point can't be extreme, because the function is about to become bigger or smaller:

    If we have a function defined on a closed interval, there will also be extreme points at the endpoints of that interval:

    Therefore we know how to find all the interesting points (that is, points that might be extreme):

    • Find all the critical points.
        
    • If looking at a function on a closed interval, toss in the endpoints of the interval.

    At this point, we have all the places where extreme points could happen. However, a critical point doesn't need to be a max or a min.

    After finding all x-values where extreme points could occur, we still need to test each x-value to see whether it really is an extreme point, and if so, what type (max or min).

    There are three ways to determine whether each possible extreme point we've found is a maximum, a minimum, or neither. Alas - the first way, while easiest, is usually not acceptable as an answer on exams. It can be a good way to check your work though.

    • Use your graphing calculator to graph the function near the possible extreme point. Then use your eyes to see what kind of point it is.
        
    • Use the First Derivative Test
        
    • Use the Second Derivative Test

    This math will come in handy for optimization, which is the art of classifying extreme points but with more word problems layered on top.

  • First Derivative Test

    Sample Problem

    1. Below is a graph of a function f with a minimum at x = x0. Determine the sign of the derivative ' at each labelled x-value.
    2. Below is a graph of a function f with a maximum at x = x0. Determine the sign of the derivative ' at each labelled x-value.

    A minimum, assuming it's not at the endpoint of an interval, usually looks like this:

    The derivative is zero (or undefined) at the place the minimum occurs:

    Since the function must decrease down to the minimum and then increase away from the minimum, the derivative is negative to the left and positive to the right of the place where the minimum occurs:

    We can use a numberline to keep track of the sign of ' like this:

    A maximum, assuming it's not at the endpoint of an interval, usually looks like this:

    The derivative is zero (or undefined) at the place the maximum occurs:

    Since the function must increase up to the maximum and then decrease away from the maximum, the derivative is positive to the left and negative to the right of the place where the maximum occurs:

    We can use a numberline to keep track of the sign of ' like this:

    If we don't have a graph of the function, we can go the other way around: we make a numberline first, and use that to determine if a critical point of f is a maximum or a minimum or neither. We find the sign of ' a little to the left of the critical point, and a little to the right of the critical point. 

    But if we encounter something like this, the critical point is neither a min nor a max:

    This process is called the First Derivative Test because we are using the first derivative to test whether a critical point is a min or a max or neither.

  • Second Derivative Test

    Sample Problems

    • Assume f is defined and twice differentiable on the whole real line. Around a minimum of the function f, is f concave up or concave down?
        
    • Assume f is defined and twice differentiable on the whole real line. Around a maximum of the function f, is f concave up or concave down?

    A minimum of f will usually occur at the bottom of a right-side up bowl:

    Having a right-side up bowl means f is concave up here.

    A maximum of f will usually occur at the top of an upside-down bowl:

    Having an upside-down bowl means f is concave down here.

    The Second Derivative Test says

    • If f is concave up around a critical point, that critical point is a minimum.
        
    • If f is concave down around a critical point, that critical point is a maximum.

    This is true because if f is concave up around a critical point, f looks like this:

    Such a critical point must be a minimum. On the other hand, if f is concave down around a critical point, then f looks like this:

    Such a critical point must be a maximum.

    Be Careful: If " is zero at a critical point, we can't use the Second Derivative Test, because we don't know the concavity of f around the critical point.

    Be Careful: There's sometimes confusion about this test because people think a concave up function should correspond to a maximum. This is why pictures are useful. If we remember what a concave up function looks like, we'll be fine.

    There's a good question that most people have right about now: if you're not told which to use, how do you know whether to use the first derivative test or the second derivative test?

    The good news is that it often doesn't matter. When it's possible to use both the first derivative test and the second derivative test, they will give the same answer.

    The other good news is that you can usually do whichever test is easier. Sometimes finding the second derivative is not fun, like with the function

    The first derivative is

    and while we could find the second derivative, it's not pretty and we don't want to bother. In this case, it probably makes more sense to plug in a couple of numbers and see what the sign of the first derivative is doing. Sometimes the second derivative test doesn't work at all (if " is 0 at the critical point), in which case we need to use the first derivative test.

    On the other hand, sometimes you can see that the second derivative is really nice. Take the function

    f (x) = x2 + 4x + 1.

    The first derivative is

    '(x) = 2x + 4

    and the second derivative is

    "(x) = 2,

    which is always positive. Therefore f is always concave up, so any critical point needs to be a minimum. The second derivative test for this one is a piece of cake. Mmm, cake.

    The bad news is that, as with the rest of math, we do need to practice. The more functions we stare at, the better we'll become at deciding whether to use the first derivative test or the second derivative test to classify a function's extreme points. Don't worry; there are plenty of practice problems.