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Rolle's Theorem Calculator

Rolle's theorem verifier. Enter f(x) on [a,b] with equal endpoints to find c with f′(c)=0, the horizontal tangent point.

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f(a)=f(b) ⇒ ∃c, f′(c)=0
Supports + - * / ^ and sin cos tan sqrt exp log pi e
f(a)=f(b)
f(a)=f(b)

📖 Tutorial｜Rolle's Theorem

1. Definition
If f is continuous on [a,b], differentiable on (a,b) and f(a)=f(b), then some c∈(a,b) has f′(c)=0 (horizontal tangent).
2. Symbols
SymbolMeaning
f(a)=f(b)Equal endpoints
cStationary point
f′(c)=0Horizontal tangent
3. How it works
  • We first check f(a)=f(b);
  • Then scan for f′≈0;
  • That is the Rolle point.
4. Steps
  1. Enter f(x);
  2. Enter an interval with f(a)=f(b);
  3. Click Calculate.
5. Example
Example: f=x²−2x on [0,2]. f(0)=f(2)=0, f′=2x−2=0 gives c=1.
6. Pitfalls
Must have f(a)=f(b);
c may not be unique;
This is the MVT with equal endpoints.

❓ FAQ｜Rolle's Theorem

Relation to the MVT?
Rolle is the MVT with f(a)=f(b), so the average slope is 0.
Why require f(a)=f(b)?
Then the chord is horizontal and the theorem asserts an interior horizontal tangent.
Can I use it if f(a)≠f(b)?
No; use the general Mean Value Theorem instead.
What is Rolle used for?
Proving f′(x)=0 has roots, or counting the number of zeros of f.
Is c always unique?
No; sin x over a period has several horizontal tangents.