Rolle's Theorem Calculator
Check the hypotheses and find every c in (a, b) with f′(c) = 0.
Enter your values
Enter the function f(x) you are working with.
Enter the interval start a you are working with.
Enter the interval end b you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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
At a glance
| What it does | Check the hypotheses and find every c in (a, b) with f′(c) = 0. |
|---|---|
| Category | Math |
| Inputs needed | Function f(x), Interval start a, Interval end b |
| Main output | Hypothesis f(a) = f(b) |
| Formula | f(a) = f(b) ⇒ ∃ c ∈ (a, b) with f′(c) = 0 |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- Function f(x)
- Enter the function f(x) you are working with.
- Interval start a
- Enter the interval start a you are working with.
- Interval end b
- Enter the interval end b you are working with.
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.
What do I need to enter into the Rolle's Theorem Calculator?
Just 3 values: function f(x), interval start a, interval end b. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Rolle's Theorem Calculator work out the 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. It applies the formula f(a) = f(b) ⇒ ∃ c ∈ (a, b) with f′(c) = 0 and shows the working so you can check each step by hand.
Is the Rolle's Theorem Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.