Rolle's Theorem

The theorem

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists at least one point $c \in (a,b)$ where $f'(c) = 0$: intuitively, if a smooth curve starts and ends at the same height, it must have a flat point somewhere in between.

Worked example

Verify Rolle's theorem for $f(x) = x^2 + 2x - 8$ on $[-4, 2]$.

Solution: $f$is a polynomial, so it's continuous and differentiable everywhere. Checking the endpoints:

Since $f(-4) = f(2)$, Rolle's theorem guarantees a point $c \in (-4, 2)$ with $f'(c)=0$. Since $f'(x) = 2x+2$:

$c=-1$ does lie in $(-4,2)$, so the theorem is verified.