Rolle's Theorem Calculator
Check the hypotheses and find every c in (a, b) with f′(c) = 0.
Enter your values
ⓘ Live results
- Hypothesis f(a) = f(b)
- Satisfied
- c value(s) with f′(c) = 0
- 2
f(1) = 0, f(3) = 0
Working
- 1Evaluate the endpoints: f(1) = 0, f(3) = 0
- 2Endpoints match, so Rolle's theorem applies.
- 3Solve f′(c) = 0 on the open interval.
Quick answer
Rolle's theorem states that if f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then there is at least one c in (a, b) with f′(c) = 0. This calculator verifies the endpoint condition and locates every such c.
f(a) = f(b) ⇒ ∃ c ∈ (a, b) with f′(c) = 0
How to use the Rolle's Theorem Calculator
- 1Enter f(x).
- 2Enter a and b with f(a) = f(b).
- 3Read the c values where the tangent is horizontal.
Worked example
Using the values the calculator loads with:
Inputs
- Function f(x)x^2 - 4x + 3
- Interval start a1
- Interval end b3
Results
- Hypothesis f(a) = f(b)Satisfied
- c value(s) with f′(c) = 02
- Evaluate the endpoints: f(1) = 0, f(3) = 0
- Endpoints match, so Rolle's theorem applies.
- Solve f′(c) = 0 on the open interval.
Frequently asked questions
What if f(a) ≠ f(b)?
Rolle's theorem does not apply. Use the mean value theorem, which replaces f′(c) = 0 with f′(c) equal to the secant slope.
Does Rolle's theorem guarantee exactly one c?
No — it guarantees at least one. A wavy function can have many horizontal tangents inside the interval.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.