Check Rolle's theorem for a polynomial on an interval: verify f(a) = f(b), find every c with f′(c) = 0, and see whether the theorem applies.
Used only for cubic polynomials.
point(s) c where f′(c) = 0
How to use this tool
Choose quadratic or cubic and type the coefficients. The default cubic x³ − 3x² on [0, 3] is a textbook example.
Type the interval endpoints with a < b. The calculator evaluates the polynomial at both ends.
Polynomials are automatically continuous and differentiable, so the only condition to check is f(a) = f(b). The checklist below shows all three.
If the theorem applies, the headline value is every c strictly inside (a, b) where the derivative is zero — the horizontal tangent the theorem guarantees.
Formula & worked example
Worked example: f(x) = x³ − 3x² on [0, 3]. f(0) = 0 and f(3) = 27 − 27 = 0, so the endpoint values match. f′(x) = 3x² − 6x = 3x(x − 2), whose roots are 0 and 2. Only 2 lies strictly inside (0, 3), so c = 2 is the guaranteed point, and indeed f′(2) = 0.
Limitations — read this before you rely on it
This calculator covers polynomials only, which are always continuous and differentiable — the interesting part of Rolle's theorem for other functions (corners, jumps, vertical tangents) is outside its scope. Derivative roots are found analytically, so no numerical root-finding error arises, but coefficients are rounded to four decimals in the display.
Frequently asked questions
If a function starts and ends at the same height on an interval, and is continuous and differentiable there, then somewhere inside the interval its tangent is horizontal — some c has f′(c) = 0. It is the special case of the mean value theorem with equal endpoint values.
Without equal endpoints there is no reason for a horizontal tangent: a steadily rising function has a positive derivative everywhere. Equal endpoints force the function to turn around somewhere, and at the turn the derivative is zero.
Yes — strictly inside (a, b), not at the endpoints. A derivative root exactly at a or b does not satisfy the theorem, and this calculator excludes endpoint roots from the headline answer.
Yes. A cubic's derivative is a quadratic with up to two roots, and both can lie inside the interval. The calculator lists every interior root.
Then Rolle's theorem does not apply — but the mean value theorem might: it guarantees a point where the derivative equals the average slope (f(b) − f(a)) ÷ (b − a) instead of zero.