Rolle’s theorem

What you are looking at

The curve is $f(x) = \dfrac{1}{2}(4 - x^2)(1 + kx)$ on $[-2, 2]$. The first factor makes the ends easy: $f(-2) = f(2) = 0$ for every $k$, so the chord $AB$ is horizontal. The theorem promises a $\xi$ in between with $f'(\xi) = 0$; the working lists all of them, and the green lines mark the horizontal tangents.

Every condition matters, and the second function shows the most common failure. $f(x) = 2 - |x|$ has equal heights at $\pm 2$ and is continuous, but it has a corner at $0$: it is not differentiable there, and no point has a horizontal tangent. Rolle’s theorem is the special case of the mean value theorem in which the chord is horizontal.

Try this

1. 1Slide $k$ from $0$ to $1$: the single $\xi = 0$ moves to the right, and once $k$ passes $0.5$ a second $\xi$ appears on the left.
2. 2Drag $P$ along the curve: the tangent is horizontal exactly at the $\xi$ in the working.
3. 3Drag $A$ or $B$ so that $f(\alpha) \ne f(\beta)$: the theorem no longer applies, although a horizontal tangent may still be there.
4. 4Switch to $2 - |x|$: equal heights and continuous, but the corner at $0$ breaks differentiability — there is no $\xi$.

Mistakes students make

• Checking $f(\alpha) = f(\beta)$ but not differentiability on $(\alpha, \beta)$: a single corner or cusp is enough to lose $\xi$.
• Asking for too much: differentiability is only needed on the open interval. $\sqrt{1 - x^2}$ on $[-1, 1]$ satisfies the theorem although it is not differentiable at $\pm 1$.
• Expecting exactly one $\xi$: the theorem gives at least one.

In the exam

• The classic use: prove an equation has at most one root (or at most $n$). If it had two, Rolle’s theorem would give a zero of $f'$ that cannot exist.
• To show there is a $\xi$ with some property, build the function: find $g$ with $g(\alpha) = g(\beta)$ whose derivative is the expression you want, and apply the theorem to $g$.

Where it is taught

• AP Calculus AB/BC · Unit 5 (5.1, the Mean Value Theorem, of which Rolle’s is the special case)
• Greek Γ΄ Λυκείου · §2.5