Rolle's Theorem: Step-by-Step Guide
Rolle's Theorem states that if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0. It is the special case of the Mean Value Theorem in which the secant slope is zero.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Verify continuity on [a, b]. Confirm f is continuous on the closed interval. Polynomials, sine, cosine, and exponentials qualify automatically; rational functions need a nonzero denominator throughout the interval.\[f \text{ continuous on } [a, b]\]
- Verify differentiability on (a, b). Confirm f is differentiable on the open interval. Watch for cusps, corners, or vertical tangents.\[f \text{ differentiable on } (a, b)\]
- Verify f(a) = f(b). Compute the values at the endpoints; they must be equal for Rolle's theorem to apply.\[f(a) = f(b)\]
- Solve f'(c) = 0 for c in (a, b). Differentiate f, set the derivative equal to zero, solve, and keep only solutions in the open interval.\[f'(c) = 0, \;\; c \in (a, b)\]
Worked example
Verify Rolle's theorem for f(x) = x^2 - 4x + 3 on [1, 3] and find c.
Step 1: f is a polynomial, so it is continuous on [1, 3].
Step 2: f is differentiable everywhere, so it is differentiable on (1, 3).
Step 3: Check the endpoints: f(1) = 1 - 4 + 3 = 0 and f(3) = 9 - 12 + 3 = 0. They are equal, so the third hypothesis holds.
Step 4: Differentiate: f'(x) = 2x - 4. Set f'(c) = 0: 2c - 4 = 0, so c = 2.
Step 5: Confirm c = 2 lies in (1, 3) — yes. Rolle's theorem is verified at c = 2.
Verification: At x = 2, f(2) = 4 - 8 + 3 = -1, the minimum of the parabola. The tangent there is horizontal, matching f'(2) = 0.
Tips and common mistakes
- Always verify ALL THREE hypotheses: continuity, differentiability, and equal endpoint values.
- A single failure of any hypothesis means Rolle's theorem does not apply — though f' may still equal 0 somewhere.
- Geometrically, between two points at the same height there must be a horizontal tangent.
- Rolle's theorem is the foundation for the Mean Value Theorem.
- If f(a) = f(b) = 0, Rolle's theorem says there is a critical point between any two consecutive zeros of f.
- Use this fact to limit the number of zeros a function can have: at most one between consecutive critical points.
- Watch for cases where f has a corner inside (a, b) — differentiability fails and the theorem may not apply.
- Always verify your c lies strictly inside the open interval — endpoints do not satisfy the conclusion.