Mathwords logoMathwords

Rolle's Theorem — Definition, Formula & Examples

Rolle's Theorem is a calculus result guaranteeing that if a function is continuous on a closed interval, differentiable on its interior, and takes equal values at both endpoints, then there is at least one point inside the interval where the derivative equals zero.

Let ff be continuous on [a,b][a, b] and differentiable on (a,b)(a, b). If f(a)=f(b)f(a) = f(b), then there exists at least one c∈(a,b)c \in (a, b) such that f′(c)=0f'(c) = 0.

Key Formula

f′(c)=0for some c∈(a,b)f'(c) = 0 \quad \text{for some } c \in (a,b)
Where:
  • ff = A function continuous on [a, b] and differentiable on (a, b) with f(a) = f(b)
  • cc = A point in the open interval (a, b) where the derivative is zero
  • a,ba, b = Endpoints of the interval

How It Works

To apply Rolle's Theorem, first verify all three hypotheses: continuity on [a,b][a, b], differentiability on (a,b)(a, b), and f(a)=f(b)f(a) = f(b). If any hypothesis fails, the theorem does not apply. Once all conditions are confirmed, you can conclude that at least one critical number cc exists in (a,b)(a, b) where the tangent line is horizontal. To find the actual value of cc, set f′(x)=0f'(x) = 0 and solve for xx within the open interval.

Worked Example

Problem: Verify Rolle's Theorem for f(x)=x2−4x+3f(x) = x^2 - 4x + 3 on [1,3][1, 3] and find the value of cc.
Check hypotheses: ff is a polynomial, so it is continuous on [1,3][1, 3] and differentiable on (1,3)(1, 3). Evaluate the endpoints:
f(1)=1−4+3=0,f(3)=9−12+3=0f(1) = 1 - 4 + 3 = 0, \quad f(3) = 9 - 12 + 3 = 0
Set the derivative equal to zero: All three conditions are satisfied, so Rolle's Theorem guarantees a cc in (1,3)(1, 3) with f′(c)=0f'(c) = 0. Compute the derivative and solve:
f′(x)=2x−4=0  ⟹  x=2f'(x) = 2x - 4 = 0 \implies x = 2
Confirm $c$ is in the interval: Since 2∈(1,3)2 \in (1, 3), the theorem is verified.
c=2c = 2
Answer: The value guaranteed by Rolle's Theorem is c=2c = 2, where f′(2)=0f'(2) = 0.

Why It Matters

Rolle's Theorem is the key lemma used to prove the Mean Value Theorem, which in turn underpins much of differential calculus. It also appears directly in proofs about the number of roots a function can have, such as showing that between two roots of ff, there must be a root of f′f'.

Common Mistakes

Mistake: Applying the theorem when f(a)≠f(b)f(a) \neq f(b) or when ff is not differentiable on the open interval.
Correction: All three hypotheses must hold. For example, f(x)=∣x∣f(x) = |x| on [−1,1][-1, 1] satisfies f(−1)=f(1)f(-1) = f(1) and is continuous, but it is not differentiable at x=0x = 0, so Rolle's Theorem does not apply.