Rolle's theorem, to move with your hands.
If a function starts and ends at the same height, somewhere in between it must have stopped rising to start falling — or the other way round. At that point the tangent is horizontal. The whole of Rolle's theorem is this sentence, and here you can check it by dragging the curve.
Let f be a function continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there exists at least one point c ∈ (a, b) such that f′(c) = 0.
Statement
The interactive figure
Use the arrows on the figure (or swipe) to build it one step at a time, then drag the two control points or pick a ready-made shape. It works from the keyboard too: ← and → change step; Tab reaches a control point, and there the arrows move it.
Hover over (or tap) a part of the drawing — the curve, the endpoints, the point c — to find out what it is.
We start from a closed interval [a, b]
The endpoints a and b are fixed and belong to the domain: the theorem requires the function to be defined and continuous exactly there too, not just “nearby”. That is why the interval is written with square brackets.
Point c: x = 0.00f′(c) = 0.00
The cleanest case: the curve is symmetric, and the point c falls exactly in the middle.
The reasoning, step by step
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Step 1 · Interval
We start from a closed interval [a, b]
The endpoints a and b are fixed and belong to the domain: the theorem requires the function to be defined and continuous exactly there too, not just “nearby”. That is why the interval is written with square brackets.
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Step 2 · Hypotheses
The two hypotheses: continuous on [a, b], differentiable in (a, b)
Continuous on the whole closed interval means the curve can be drawn without lifting the pen. Differentiable on the open interval means that at every interior point the tangent exists: no corners, no points with a vertical tangent. At the endpoints the derivative is not needed, which is why the second interval is open.
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Step 3 · f(a) = f(b)
The third hypothesis: the function returns to its starting point
f(a) and f(b) are the same number, so the segment joining the two endpoints is horizontal: its slope is (f(b) − f(a)) / (b − a) = 0. Keep this zero in mind, because it is exactly the number we will find again at the end.
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Step 4 · c exists
The conclusion: there is at least one interior point c
If the curve goes up, sooner or later it has to come back down to find its starting height again — or the other way round. At the point where it turns round the function has an interior maximum or minimum, and at an interior maximum or minimum of a differentiable function the derivative is zero. That point is c.
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Step 5 · Tangent
At c the tangent is horizontal: f′(c) = 0
The tangent drawn at c is parallel to the x axis, that is it has the same slope as the segment between the endpoints. This is exactly what Rolle says: somewhere inside the interval the instantaneous slope coincides with the average one, which here is zero.
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Step 6 · Try it
Now try to break the theorem
Drag the two control points or pick one of the ready-made shapes: the curve changes, the point c moves, sometimes it becomes two — but you will not be able to make it disappear. As long as f(a) = f(b) and the curve stays smooth, there is always a point with a horizontal tangent. That is the meaning of the word “at least” in the statement.
What each element of the drawing is
In the figure these explanations appear when you hover over the element. Here they are all written out.
- The interval [a, b]
- The domain we are reasoning on, closed — the square brackets include a and b. The function must be defined and continuous exactly there too, not just at the “nearby” points.
- The endpoints A and B
- The points (a, f(a)) and (b, f(b)). They are at the same height: this is the hypothesis f(a) = f(b), and in the drawing it cannot be violated because the curve is built to pass through there.
- The line f(a) = f(b)
- The horizontal segment joining the two endpoints. Its slope is zero: it is the average slope of the function over the interval.
- The function f
- Continuous on [a, b] and differentiable in (a, b): no corners, no jumps, no points with a vertical tangent. Without one of these properties the theorem falls.
- The point c
- An interior point of the interval where the derivative is zero. It is not found by eye: it is the exact solution of the equation f′(x) = 0, recalculated every time you move the curve.
- The tangent at c
- The line tangent to the graph at the point c. Since f′(c) = 0 it is horizontal, so parallel to the line joining the endpoints.
- The control points
- Drag them up and down (or use the keyboard arrows) to change the shape of the function. The endpoints stay put, so the hypothesis f(a) = f(b) remains true whatever you do.
Why the proof works
A function that is continuous on a closed, bounded interval attains an absolute maximum and minimum on that interval: this is Weierstrass's extreme value theorem, and the closedness of [a, b] is needed exactly for this. There are then two cases. If the maximum and the minimum both occur at the endpoints, they are the same number — because f(a) = f(b) — so the function is constant and the derivative is zero everywhere. Otherwise at least one of them occurs at an interior point c.
At that interior point Fermat's theorem comes in: at an interior maximum or minimum of a differentiable function the derivative is zero, because if it were positive the function would still be rising and if it were negative it would be falling — either way it would not be an extremum. So f′(c) = 0. That is all there is to it: Weierstrass guarantees that an extremum exists, Fermat says the derivative is zero there, and f(a) = f(b) is the hypothesis that stops the extremum hiding at the edges.
Note which hypothesis is needed where: continuity is used by Weierstrass, and it needs it at the endpoints too; differentiability is used by Fermat, and it only needs it in the interior. It is not a detail of notation — it is the reason the first interval is closed and the second is open.
The questions we always get.
Does Rolle's theorem say that c is unique?
No. It says there is at least one point c with f′(c) = 0. There can be two, ten or infinitely many: in the visualisation the “Two points c” shape shows two, and if the function is constant every point of the interval works.
Why is [a, b] closed and (a, b) open?
Continuity is needed at the endpoints too, otherwise the function could “escape” exactly at a or b and have no maximum or minimum. Differentiability at the endpoints is not needed: the point c the theorem guarantees is interior, so it is enough for the tangent to exist inside.
How are Rolle's theorem and Lagrange's theorem related?
Rolle is the special case of Lagrange with f(a) = f(b). Lagrange (the mean value theorem) says there is a c where the instantaneous slope equals the average one (f(b) − f(a)) / (b − a); when the two endpoint values coincide that average slope is zero, and you recover exactly Rolle's statement.
What happens if one of the hypotheses fails?
The theorem no longer guarantees anything. With f(x) = |x| on [−1, 1] we have f(−1) = f(1) but at 0 there is a corner: the function is not differentiable there and no other point has a zero derivative. A single point where the function is not differentiable is enough to bring the conclusion down.
Keep going from here
Calculus that will not sink in?
Rolle, Lagrange, Cauchy, L'Hôpital: four statements that lean on one another, and anyone who learns them by heart in no particular order mixes them up in every test. Let us start again from the first one that does not add up for you.