Source: Stewart, Calculus 7th Ed., Ch. 3.2 | Math 132, MSU
Tags: Rolle's Theorem, vanishing derivatives, critical points, continuous, differentiable
Difficulty: Intermediate | Prerequisites: Limits and continuity (Ch. 1-2), definition of the derivative, Extremal Value Theorem (Ch. 3.1), First Derivative Theorem (Ch. 3.1)
Rolle's Theorem is the first of several results in Chapter 3.2 that connect a function's derivative to its values. It sits between the Extremal Value Theorem (which guarantees max/min points exist) and the Mean Value Theorem (which generalises Rolle's result). The idea is simple: if a continuous, differentiable curve starts and ends at the same height, it must have a horizontal tangent somewhere in between. Everything in this section, including the MVT and the Uniqueness Theorem, builds on Rolle's, so getting comfortable with its hypotheses and logic is essential before moving forward.
If a smooth curve starts and ends at the same y-value on a closed interval, the derivative must equal zero at least once in the interior. Rolle's Theorem is the foundation for the Mean Value Theorem, which in turn underpins uniqueness results for differential equations and physical laws.
Closed interval [a, b]
The set of all x satisfying a <= x <= b, including both endpoints.
Think of it as the full stretch of road from point a to point b, fences included.
Open interval (a, b)
The set of all x satisfying a < x < b, excluding the endpoints.
In simple terms, this means everything between a and b, but not a or b themselves.
Continuous on [a, b]
The function has no breaks, jumps or holes anywhere on the closed interval. You can draw it without lifting your pen.
Differentiable on (a, b)
The function has a well-defined derivative (no corners, cusps or vertical tangents) at every point in the open interior. The endpoints are deliberately excluded, so the function is allowed to misbehave at a or b.
Hypothesis (plural: hypotheses)
The "if" conditions of a theorem, the setup you must verify before using the conclusion. For Rolle's Theorem there are three: continuity on [a, b], differentiability on (a, b), and f(a) = f(b).
Conclusion
The "then" payoff of a theorem, the guaranteed result once all hypotheses hold. For Rolle's Theorem: there exists some c in (a, b) with f'(c) = 0.
Q.E.D.
Abbreviation of the Latin quod erat demonstrandum, meaning "which was to be shown." Placed at the end of a proof to mark its completion.
If f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) where f'(c) = 0.
Geometrically: the curve must be horizontal (have a flat tangent) somewhere between a and b.
Physically: if an object starts and finishes at the same height over a time interval, there must be a moment where it pauses (velocity = 0) before doubling back.
Continuity on [a, b]: without it, the function can jump over values where f' might have been zero. The Extremal Value Theorem (which the proof relies on) requires continuity.
Differentiability on (a, b): the counterexample y = g(x) with a corner at x = 1 shows why. The function g(x) has g(0) = g(2) = 0 but g'(c) is never zero, because g'(1) does not exist. The velocity jumps from 1 to -1 instantaneously, like an idealised ping-pong ball at the moment of impact.
f(a) = f(b): without equal endpoint values, the function could simply climb from one value to another without ever levelling off.
All three conditions must hold simultaneously. One failure at a single point can break the conclusion.
The proof uses the Extremal Value Theorem and the First Derivative Theorem from Section 3.1.
Case 1: The absolute maximum c_1 lies in the interior (a, b). The First Derivative Theorem gives f'(c_1) = 0.
Case 2: The absolute maximum sits at an endpoint (c_1 = a or b), so f(c_1) = f(a) = f(b). Look at the absolute minimum c_2 instead. If c_2 is in the interior, then f'(c_2) = 0.
Case 3: Both the maximum and minimum occur at the endpoints. Since f(a) = f(b), the max and min values are equal, meaning f(x) is constant on [a, b]. A constant function has f'(c) = 0 for every c in (a, b).
In every case, the conclusion holds.
f'(c) = 0 \quad \text{for some } c \in (a, b)Conditions:
f \text{ continuous on } [a, b], \quad f \text{ differentiable on } (a, b), \quad f(a) = f(b)If you drive from home and return to exactly the same spot, your speedometer must have read zero at least once during the trip (you stopped or reversed direction). Rolle's Theorem is the mathematical formalisation of that observation. It also underpins the Mean Value Theorem, which engineers and physicists use whenever they relate average and instantaneous rates of change.
Students often think that f'(c) = 0 means there is only one such point c. Rolle's Theorem guarantees at least one, but there can be several.
Students sometimes confuse [a, b] and (a, b). The function must be continuous on the closed interval (endpoints included) but only needs to be differentiable on the open interval (endpoints excluded).
A common error is applying Rolle's Theorem without checking f(a) = f(b). If the endpoint values differ, the theorem does not apply, and you need the Mean Value Theorem instead.
Some students believe that if the conclusion fails, all three hypotheses must fail. In fact, dropping just one hypothesis at just one point can invalidate the conclusion.
Rolle's Theorem is the stepping stone to the Mean Value Theorem. Exam problems frequently ask you to verify all three hypotheses before applying either result.
Watch for problems where differentiability fails at a single interior point (corners, cusps, vertical tangents). These are classic "does Rolle's apply?" exam questions.
Proof-based exams may ask you to reproduce the three-case structure of the proof.
True or False: Rolle's Theorem requires f(a) = f(b). ___
True or False: f(x) must be differentiable at the endpoints a and b. ___
Fill in the blank: The conclusion of Rolle's Theorem guarantees that f'(c) = ___ for some c in (a, b).
True or False: If f(x) has a corner at a single point inside (a, b), Rolle's Theorem still applies. ___
True or False: Rolle's Theorem guarantees exactly one value of c. ___
Answers: 1. True. 2. False (only on the open interior). 3. Zero. 4. False (differentiability fails). 5. False (at least one, possibly more).
Q: State the three hypotheses of Rolle's Theorem.
A: (1) f is continuous on the closed interval [a, b]. (2) f is differentiable on the open interval (a, b). (3) f(a) = f(b).
Q: Let f(x) = x^2 - 4x + 3 on the interval [1, 3]. Verify the hypotheses of Rolle's Theorem and find the value of c.
A: f is a polynomial, so it is continuous and differentiable everywhere. f(1) = 1 - 4 + 3 = 0 and f(3) = 9 - 12 + 3 = 0, so f(a) = f(b). All hypotheses hold. Setting f'(x) = 2x - 4 = 0 gives c = 2, which lies in (1, 3).
Q: Give an example of a function on [0, 2] where f(0) = f(2) but Rolle's Theorem does not apply. Explain why.
A: f(x) = |x - 1|. This function has f(0) = 1 = f(2) and is continuous on [0, 2], but it is not differentiable at x = 1 (a corner). The derivative jumps from -1 to 1 at that point, and f'(c) is never zero on (0, 2).
Q: Why does the proof of Rolle's Theorem need to consider three separate cases?
A: The Extremal Value Theorem guarantees a maximum exists on [a, b], but that maximum might sit at an endpoint rather than the interior. Case 1 handles an interior maximum. Case 2 handles an endpoint maximum but interior minimum. Case 3 handles both extremes at the endpoints, forcing f to be constant.
Rolle's Theorem is a special case of the Mean Value Theorem (see the MVT study notes): set f(a) = f(b) and the MVT conclusion reduces to f'(c) = 0. The MVT proof actually constructs an auxiliary function g(x) specifically so that Rolle's Theorem applies to it. This chain, from Rolle's to MVT to the Uniqueness Theorem, is how the course proves that a differential equation with an initial condition has exactly one solution, which is the mathematical reason physical laws give unique predictions.
Rolle's Theorem, vanishing derivative, horizontal tangent, critical point, f'(c) = 0, continuous function, differentiable function, closed interval, open interval, Extremal Value Theorem, First Derivative Theorem, Mean Value Theorem, MVT, intermediate value, calculus proof, Stewart 3.2, Math 132, Calculus I, MSU