Rolle's theorem

What you will be able to do

• state Rolle's theorem and check its three conditions;
• find the point $c$ for a given function;
• use the theorem to count the roots of an equation.

The theorem

Walk along a hilly road from a point at height $h$ to another point at the same height $h$. Somewhere you were at the top of a hill or at the bottom of a valley. There the road was level. This is Rolle's theorem.

Let $f$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $f(a) = f(b)$. Then there is at least one number $c$ in $(a, b)$ with $f'(c) = 0$.

The idea of the proof is short. A continuous function on a closed interval has a greatest and a least value. If they are both equal to $f(a)$, the function is constant, and $f' = 0$ everywhere. Otherwise one of them is reached at a point $c$ inside, and at such a point the slope must be zero.

Why each condition is needed

• Differentiability. $f(x) = |x|$ on $[-1, 1]$ has equal ends, but a corner at $0$. Its slope is never zero.
• Equal ends. $f(x) = x$ on $[0, 1]$ has no horizontal tangent, because $f(0) \ne f(1)$.
• Continuity. Let $f(x) = x$ for $0 \le x < 1$ and $f(1) = 0$. The ends are equal, but the jump at $1$ breaks the theorem.

Finding the point $c$

Check the three conditions. Then solve $f'(c) = 0$ and keep the solutions that lie in the open interval $(a, b)$.

• For $f(x) = x^2 - 4x + 3$ on $[1, 3]$: $f(1) = 0 = f(3)$. Then $f'(x) = 2x - 4 = 0$ gives $c = 2$.
• For $f(x) = \sin x$ on $[0, \pi]$: $f(0) = 0 = f(\pi)$. Then $\cos c = 0$ gives $c = \dfrac{\pi}{2}$.

Using the theorem to count roots

Rolle's theorem turns a statement about roots of $f$ into one about roots of $f'$.

• Between two roots of a differentiable function there is a root of its derivative.
• So if $f'$ has no real root, $f$ has at most one real root.
• If $f'$ has at most $k$ roots, $f$ has at most $k + 1$ roots.

The Intermediate Value Theorem from the lessons on continuity then shows that a root exists. Together the two theorems say how many roots an equation has.

For example, let $f(x) = x^3 + 3x + 2$. Then $f'(x) = 3x^2 + 3$ is never zero. So $f$ has at most one real root. Also $f(-1) = -2 < 0$ and $f(0) = 2 > 0$, so a root exists. Hence $f$ has exactly one real root.

A note on syllabuses. CBSE and JEE ask Rolle's theorem directly. A-Level and IB do not name it. AP Calculus treats it as a special case of the Mean Value Theorem.

A worked example

Verify Rolle's theorem for $f(x) = \sin 2x$ on $\left[0, \dfrac{\pi}{2}\right]$ and find $c$.

1. Continuity and differentiability. $f$ is a sine of a linear function, so it is continuous and differentiable everywhere.
2. Equal ends. $f(0) = 0$ and $f\left(\dfrac{\pi}{2}\right) = \sin\pi = 0$.
3. Solve $f'(c) = 0$. $f'(x) = 2\cos 2x$. It is zero when $2x = \dfrac{\pi}{2}$, so $c = \dfrac{\pi}{4}$.
4. Check the interval. $\dfrac{\pi}{4}$ lies in $\left(0, \dfrac{\pi}{2}\right)$.

Practise with new numbers

Try it in Python

The program takes $f(x) = (x - 1)(x - 4)(x - 6)$, which is zero at $1$, $4$ and $6$. By Rolle's theorem, $f'$ has a zero in each gap. The program finds them by halving the interval, as in the lesson on the Intermediate Value Theorem.

Common mistakes

• Skipping a condition. Always check continuity, differentiability and equal ends.
• Giving the end point. The point $c$ must lie in the open interval $(a, b)$.
• Taking the theorem for a converse. A horizontal tangent can exist with no equal ends. Rolle's theorem gives a sufficient condition only.
• Forgetting the existence. The theorem shows that $c$ exists. It does not say that $c$ is the only one.

Remember these

• Rolle: $f$ continuous on $[a, b]$, differentiable on $(a, b)$, $f(a) = f(b)$ implies $f'(c) = 0$ for some $c$ in $(a, b)$.
• Geometric meaning: a horizontal tangent between two points of equal height.
• Between two roots of $f$ there is a root of $f'$.
• Rolle with the Intermediate Value Theorem counts the roots of an equation.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Does Rolle's theorem apply to $f(x) = x^2 - 6x + 5$ on $[1, 5]$? If so, find $c$.

2. Why does Rolle's theorem not apply to $f(x) = |x|$ on $[-1, 1]$?

3. Does Rolle's theorem apply to $f(x) = x^2$ on $[0, 2]$?

4. Find $c$ for $f(x) = \sin 2x$ on $\left[0, \dfrac{\pi}{2}\right]$.

5. A polynomial $p$ has roots at $1$, $4$ and $7$. How many roots must $p'$ have in $(1, 7)$?

References

• OpenStax, Calculus Volume 1, Chapter 4, section on the Mean Value Theorem (Rolle's theorem is the first part).
• NCERT, Mathematics Part I for Class XII, Chapter 5, section on the Mean Value Theorem.