What Rolle's Theorem Is Really Saying
Rolle's Theorem is one of the simplest and most useful consequences of the Mean Value Theorem (MVT). It applies when a function has the same value at both endpoints of an interval.
Imagine a smooth curve that starts at one height and finishes at exactly the same height. Somewhere between those two endpoints, the curve must have a horizontal tangent. In other words, there must be at least one point where the derivative equals zero.
That is the central idea:
Rolle's Theorem does not tell you that there is exactly one such point. It guarantees that at least one point exists.
The Three Conditions
Before using Rolle's Theorem, you must verify three conditions. These are essentially the same first two conditions used by the Mean Value Theorem, plus the special requirement that the endpoint values are equal.
- Continuous on the closed interval. The function must be continuous on .
- Differentiable on the open interval. The function must be differentiable on .
- Equal endpoint values. The function must satisfy .
If all three conditions are satisfied, Rolle's Theorem guarantees that there is at least one number such that:
Why Rolle's Theorem Is a Special Case of the MVT
The Mean Value Theorem says that if a function is continuous on and differentiable on , then there is some such that:
But Rolle's Theorem adds the condition . That makes the secant slope on the left side equal to zero:
So the MVT becomes:
That is exactly Rolle's Theorem. Rolle's Theorem is therefore the Mean Value Theorem when the two endpoint values are equal.
A Geometric Interpretation
The MVT says that somewhere on a smooth curve, the instantaneous slope must equal the average slope between the endpoints.
If the endpoints have the same height, the average slope is zero. Therefore, somewhere in between, the instantaneous slope must also be zero.
Worked Examples
Show that Rolle's Theorem applies to on , and find all values of guaranteed by the theorem.
Verify that Rolle's Theorem applies to on , then find the value of guaranteed by the theorem.
Find the value of so that Rolle's Theorem applies to on .
Determine whether Rolle's Theorem applies to on .
Rolle's Theorem vs. the Mean Value Theorem
The two theorems are almost identical. The easiest way to distinguish them is to focus on the endpoint condition and the conclusion.
For the MVT, the derivative equals the average rate of change. For Rolle's Theorem, that average rate of change happens to be zero because the endpoint values are equal.
How to Approach an AP Calculus Problem
- Write down the interval. Identify and .
- Check continuity on the closed interval. Look for holes, jumps, vertical asymptotes, or other discontinuities.
- Check differentiability on the open interval. Look for corners, cusps, discontinuities, or vertical tangents.
- Check the endpoint values. Calculate and .
- If all conditions hold, write the theorem's conclusion. There exists at least one such that .
- If the problem asks for the point(s), solve .
Common Mistakes
A function can be continuous and differentiable and still fail to satisfy Rolle's Theorem. You must verify . This is the special condition that distinguishes Rolle's Theorem from the general MVT.
Rolle's Theorem requires continuity on the closed interval but differentiability only on the open interval. The conditions are and .
Rolle's Theorem guarantees at least one point. A function can have several horizontal tangents inside the interval.
The conclusion of a theorem is only guaranteed when its conditions are satisfied. On AP free-response questions, explicitly state why the function is continuous, why it is differentiable, and why the endpoint values are equal.
Practice Problems
Verify that Rolle's Theorem applies to on , and find all values of guaranteed by the theorem.
Show answer
Does Rolle's Theorem apply to on ? Explain.
Show answer
Find the value of so that Rolle's Theorem applies to on .
Show answer
Explain why Rolle's Theorem is a special case of the Mean Value Theorem when .
Show answer
Test yourself: AP Calculus practice exams
Ready to check what you know? Work an AP-style practice exam, then open the full solutions.
Want all 1,114 practice problems?
This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.
Get the Calc Pass App
Scan to download and practice on the go, with instant scoring.
Vol. 1: 1,114 Questions
The complete practice question bank, organized by AB & BC topic.
View on Amazon