5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points

The Extreme Value Theorem says that if a function is continuous on a closed interval $[a,b]$, it must have both an absolute maximum and an absolute minimum on that interval. Critical points are where the first derivative equals zero or fails to exist, and while all local extrema happen at critical points, not every critical point is an extreme value. For AP Calculus, check endpoints and critical points when finding absolute extrema.

Why This Matters for the AP Calculus Exam

This topic sets up almost everything else in Unit 5. Once you can find critical points and confirm when extrema are guaranteed, you can move into the first derivative test, the candidates test, and optimization. On the AP Calculus exam, you will see these ideas in multiple-choice questions and in free-response questions that ask you to justify conclusions about a function's behavior. Clear justification language, like naming f, f', and f'' instead of saying "it," is important for clean exam work.

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Key Takeaways

• The Extreme Value Theorem guarantees an absolute max and absolute min, but only when f is continuous on a closed interval [a, b].
• A critical point occurs where f'(x) = 0 or where f'(x) does not exist.
• All local (relative) extrema occur at critical points, but not all critical points are extrema.
• Global (absolute) extrema are the highest and lowest values over the whole interval; local (relative) extrema are highest or lowest compared to nearby points.
• On a closed interval, always check endpoints too, since an absolute extremum can occur there.
• If you are not told a function is continuous, you cannot assume the Extreme Value Theorem applies.

Extreme Value Theorem

A function $f$ defined on a closed interval $[a,b]$ must have both a maximum and a minimum value within that interval. This is the Extreme Value Theorem, and it holds as long as the function is continuous over that interval. If you want a refresher on continuity, check this guide: Confirming Continuity over an Interval.

Look at the graph below. On the interval shown, there is a minimum value, a maximum value, and possible extreme values at the endpoints. It is fine for the minimum or maximum value to occur at an endpoint. In fact, that is expected, and it is something you always need to check.

One way to picture this: think of the function as a roller coaster. As long as the track is continuous with no breaks or gaps (no discontinuities), you can expect it to have both a highest peak (maximum) and a lowest dip (minimum) somewhere along the ride.

The catch is the word "continuous." If the function has a break, jump, or hole on the interval, the theorem does not promise an absolute max or min.

Global Versus Local Extrema

Global extrema are the absolute maximum and minimum values of a function over its entire domain or interval. These are the absolute highest and lowest points when you look at the whole function.

Local extrema focus on specific regions within the function. These points might not be the absolute highest or lowest overall, but they are peaks and valleys compared to the points right around them. A point is a local maximum or minimum if it is the highest or lowest value relative to the values directly surrounding it.

Take a look at the image below and try to label each extreme value as local or absolute and as a maximum or minimum on the interval shown.

Because the function is continuous, the Extreme Value Theorem tells us there must be one absolute maximum and one absolute minimum on the interval. There are also several other points that count as local extrema because they are higher or lower than the points around them. Ask yourself whether any of those local points are also global extrema.

Now check your answers against the labeled version:

Critical Points

Critical points are where extrema can occur. A critical point is a value $x = c$ in the domain of $f$ where the function is either not differentiable or its derivative equals zero: $f'(c) = 0$.

Think of critical points as potential turning points on the roller coaster. At these spots, the track may pause or change direction. These are the candidates for local maxima and minima.

Look at the graph below. All of its extrema are critical points, but not all critical points are extrema. Point $P$ is a critical point because the derivative there is 0, but it is not a local maximum or minimum.

To find critical points, look for where:

• $f'(x) = 0$
• $f'(x)$ is undefined (for example, a cusp, corner, or vertical tangent)

After you find them, you can test each critical point to decide whether it is a local max, a local min, or neither. That comes in a later topic.

How to Use This on the AP Calculus Exam

MCQ

• Read whether the function is described as continuous and whether the interval is closed. The Extreme Value Theorem only applies when both are true.
• When given a graph of $f'(x)$, find where it crosses zero or is undefined to locate critical points of $f$.
• Remember a critical point is not automatically an extreme value. Confirm with the surrounding behavior.

Free Response

• Justify conclusions clearly. State the condition you are using, like continuity on a closed interval, before claiming a guaranteed max or min.
• Refer to $f$, $f'$, and $f''$ by name instead of "it" so your reasoning is unambiguous.
• When asked for absolute extrema on a closed interval, check both the critical points and the endpoints.

Common Trap

If a question only says a function is defined on an interval but does not say it is continuous, you cannot apply the Extreme Value Theorem. In that case, an absolute max or min is not guaranteed.

Practice Problems

1) Identifying Critical Points from a Graph

Given the graph of $f'(x)$, which points of $f(x)$ are critical points on the interval $[0,7]$?

At a critical point, the derivative either equals $0$ or does not exist. So check the graph of $f'(x)$ to see where it equals $0$. Since $f'(x) = 0$ at $x = 0$ and $x = 4$, there are critical points at $x = 0$ and $x = 4$.

2) Identifying Extrema from a Graph

Given the graph of $f(x)=x^{4}-4x^{3}+4x^{2}$, identify whether all critical points qualify as extrema, and find an absolute maximum and minimum on the interval $[-1, 2.5]$.

Start with the critical points. At $(0,0)$ there is a local minimum because all surrounding points are greater. The same is true at $(2,0)$. Because these two points are also the lowest values on the interval, $(0,0)$ and $(2,0)$ are absolute minimums.

The critical point $(1,0)$ is a local maximum, since all of its surrounding values are lower.

Now look at the endpoints. The point $(-1,9)$ is the absolute maximum on the interval $(-1, 2.5)$ because it is higher than every other point in the interval. The point $(2.5, 1.5625)$ is only a local maximum, since it is higher than the points near it but not the highest overall.

3) Extreme Value Theorem

A function $f(x)$ is defined on the interval $(3,9)$. Is the function guaranteed to have a maximum and minimum value on this interval?

To apply the Extreme Value Theorem, the function must be continuous on a closed interval. Here, you are not told that $f(x)$ is continuous, and you are not given an equation to check. So $f(x)$ is not guaranteed a maximum and minimum value on this interval.

Common Misconceptions

• "Every critical point is a max or min." Not true. A critical point is only a candidate. Point $P$ above has $f'(P) = 0$ but is neither a max nor a min.
• "The Extreme Value Theorem applies to any function." It only applies when $f$ is continuous on a closed interval $[a, b]$. Drop either condition and the guarantee disappears.
• "Critical points only happen where the derivative is zero." They also occur where the derivative does not exist, such as at a cusp, corner, or vertical tangent.
• "Endpoints do not matter." On a closed interval, absolute extrema can occur at the endpoints, so always check them.
• "Local and global mean the same thing." A local extremum is highest or lowest only compared to nearby points, while a global extremum is the highest or lowest over the whole interval.

Related AP Calculus Guides

• Unit 5 Overview: Analytical Applications of Differentiation
• 5.1 Using the Mean Value Theorem
• 5.3 Determining Intervals on Which a Function is Increasing or Decreasing
• 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
• 5.11 Solving Optimization Problems
• 5.10 Introduction to Optimization Problems