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

Abigail Young

7 min read

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Study Guide Overview

This study guide covers the Extreme Value Theorem, differentiating between global and local extrema, and identifying critical points. It explains how to apply the Extreme Value Theorem to continuous functions on closed intervals. The guide also discusses finding critical points by checking where the derivative is zero or undefined. Practice problems involving identifying critical points and extrema from graphs, and applying the Extreme Value Theorem are included.

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

Welcome back to AP Calculus with Fiveable! This topic focuses on extrema in an interval. Let’s dive right into the world of extreme values, both global and local, and the crucial concept of critical points. 🙌

#🎢 Extreme Value Theorem

Let's start with the Extreme Value Theorem. The College Board AP Calculus Exam description states that a function $f$ defined on a closed interval $[a,b]$ must have both a maximum and minimum value within that interval. This is known as the Extreme Value Theorem, and it holds true if the function is continuous over the given interval $(a,b)$ . Check out this Fiveable guide to review continuity: Confirming Continuity over an Interval.

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Graph of function with max and min value

Graph created with Virtual Graph Paper

To apply this theorem effectively, think of the function as a roller coaster. As long as the roller coaster is continuous without any breaks or disruptions (discontinuities), you can expect it to have both a highest peak (maximum) and a lowest dip (minimum) somewhere along the ride. 🎢

#🌐 Global Versus Local Extrema

Now, let's explore the difference between global and local extrema. Global extrema are the absolute maximum and minimum values of a function over its entire domain. We can identify these points because they are the absolute highest or lowest points when considering the function as a whole.

📍 Local extrema, on the other hand, focus on specific regions or intervals within the function. Thes...

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Question 1 of 11

A continuous function $f(x)$ is defined on the closed interval $[a, b]$ . According to the Extreme Value Theorem, what is guaranteed?

The function has at least one critical point

The function has both a global maximum and a global minimum within the interval

The function is differentiable on the interval

The function has a local maximum at every critical point