Rolle's theorem

Rolle’s theorem relates the values a function takes at the endpoints of an interval to the existence of a point with a horizontal tangent in its interior.

Theorem — Rolle

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists at least one point $c\in(a,b)$ such that $f'(c)=0$.

Remark — Geometric meaning

If the function starts and ends at the same height ($f(a)=f(b)$), somewhere in between it must have a horizontal tangent. It is “obvious” when looking at the graph, but the proof requires Weierstrass’s theorem.

Curve with $f(a)=f(b)$: at the point $c$ (where $x\approx 2{,}5$) the tangent is horizontal, that is $f'(c)=0$.

Example — Checking the hypotheses

Consider $f(x) = x^2-4x+3$ on $[1,3]$.

Checking the hypotheses: $f$ is a polynomial, hence continuous and differentiable everywhere ✓. Moreover $f(1) = 1-4+3 = 0$ and $f(3) = 9-12+3 = 0$, so $f(a)=f(b)$ ✓.

Finding $c$: we impose $f'(x) = 2x-4 = 0 \iff x = 2$. Check: $c=2\in(1,3)$ ✓.

Meaning: at the point $x=2$ the graph has a horizontal tangent (it is the vertex of the parabola).

Explore the theorem in the simulation: move the test point along the curve and look for where the tangent becomes horizontal.

Drag the slider $x$: the tangent to the graph becomes horizontal at $c=3$, where $f'(c)=0$.

Links

Topics: Theorems of calculus
Concepts: Derivative · Tangent · Rolle’s theorem
Methods: Rolle Lagrange
People: Michel Rolle