Rolle Theorem Calculator
Work out rolle theorem instantly with clear inputs, formula shown and shareable results.
About Rolle Theorem Calculator
Rolle's theorem is the special case of the mean value theorem where f(a) = f(b). Then the secant is horizontal, so some interior point must have a horizontal tangent.
Formula
Rolle's theorem
if f(a) = f(b) then f'(c) = 0 for some c in (a, b)
Worked example
With the default values below, Rolle Theorem Calculator returns Does Rolle's hypothesis hold of Yes, f(a) = f(b) = 0, so Rolle's theorem applies. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Polynomial coefficients, highest power first
- 1, -5, 6, 0
- Interval start a
- 2
- Interval end b
- 3
Results
- Does Rolle's hypothesis hold
- Yes, f(a) = f(b) = 0, so Rolle's theorem applies
- Point c with f'(c) = 0
- 2.54858377
- f(a)
- 0.000000
- f(b)
- 0.000000
Step by step
- 1Values used
- 2Rolle's theorem
- 3Does Rolle's hypothesis hold= Yes, f(a) = f(b) = 0, so Rolle's theorem applies
- 4Point c with f'(c) = 0= 2.54858377
- 5f(a)= 0.000000
- 6f(b)= 0.000000
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
For x³ − 5x² + 6x on [2, 3]?
The polynomial factors as x(x − 2)(x − 3), so f(2) = f(3) = 0 and Rolle applies; f' = 3x² − 10x + 6 vanishes at x ≈ 2.5486.
Why does it need differentiability?
Because |x| on [−1, 1] has f(−1) = f(1) but no horizontal tangent — the corner at zero breaks the theorem.