Mean value theorems

Rolle's theorem and Lagrange's Mean Value Theorem (MVT) are existence theorems: they tell us a point with a certain property must exist, without telling us where. Despite the modest statements, they are the engine of nearly every proof in single-variable calculus , monotonicity, Taylor's theorem, L'Hôpital's rule.

Rolle's theorem

Theorem. If $f$ is

1. continuous on $[a, b]$,
2. differentiable on $(a, b)$, and
3. $f(a) = f(b)$,

then there exists at least one $c \in (a, b)$ such that $f'(c) = 0$.

Geometric meaning. If a smooth curve starts and ends at the same height, it must have a horizontal tangent somewhere in between.

Lagrange's Mean Value Theorem

Theorem. If $f$ is

1. continuous on $[a, b]$ and
2. differentiable on $(a, b)$,

then there exists at least one $c \in (a, b)$ such that

$f'(c) = \frac{f(b) - f(a)}{b - a}.$

Geometric meaning. The slope of the chord from $(a, f(a))$ to $(b, f(b))$ equals the slope of the tangent at some interior point $c$.

Proof of Lagrange's MVT from Rolle

Define $g(x) = f(x) - L(x)$, where $L(x)$ is the line through $(a, f(a))$ and $(b, f(b))$. Then $g(a) = g(b) = 0$, so by Rolle's theorem there is $c$ with $g'(c) = 0$, i.e., $f'(c) = L'(x) = \dfrac{f(b) - f(a)}{b - a}$. $\blacksquare$

Why the hypotheses matter

Rolle's theorem can fail if:

• $f$ is not continuous on $[a, b]$: take $f$ with a jump.
• $f$ is not differentiable on $(a, b)$: e.g., $f(x) = |x|$ on $[-1, 1]$.
• $f(a) \neq f(b)$.

Similar comments for Lagrange.

Consequences

Monotonicity: If $f'(x) > 0$ on an interval, then $f$ is strictly increasing on it. Reason: for any $a < b$ in the interval, MVT gives $f(b) - f(a) = f'(c)(b - a) > 0$.

Constants: If $f'(x) = 0$ on an interval, then $f$ is constant.

Bounded slopes: If $|f'(x)| \le M$ on an interval, then $|f(b) - f(a)| \le M|b - a|$ , Lipschitz condition.

Worked example

Verify Rolle's theorem for $f(x) = x^2 - 4x + 3$ on $[1, 3]$.

$f(1) = 0, f(3) = 0$. Continuous and differentiable everywhere. By Rolle, $\exists c \in (1, 3)$ with $f'(c) = 0$. $f'(c) = 2c - 4 = 0 \Rightarrow c = 2$. $\checkmark$

Verifying Lagrange's MVT

For $f(x) = x^3$ on $[1, 4]$:

Slope of chord = $\dfrac{64 - 1}{3} = 21$. MVT promises $c \in (1, 4)$ with $f'(c) = 21$, i.e., $3 c^2 = 21$, so $c = \sqrt{7} \approx 2.65 \in (1, 4)$. $\checkmark$

Worked examples

Example 1. Verify Rolle for $f(x) = \sin x$ on $[0, \pi]$.

$f(0) = f(\pi) = 0$. Continuous, differentiable. $f'(c) = \cos c = 0 \Rightarrow c = \pi/2 \in (0, \pi)$. $\checkmark$

Example 2. Use MVT to show $\sin a - \sin b \le |a - b|$ for all reals $a, b$.

By MVT, $\sin a - \sin b = (a - b) \cos c$ for some $c$ between. Since $|\cos c| \le 1$, $|\sin a - \sin b| \le |a - b|$. $\blacksquare$

Example 3. Show $f(x) = x^3 - 3x + 2$ has at most one root in $(-1, 1)$.

Compute $f'(x) = 3x^2 - 3 = 3(x^2 - 1)$. For $x \in (-1, 1)$, $f'(x) < 0$, so $f$ is strictly decreasing. A strictly monotonic function has at most one zero.

Example 4. Apply Rolle to $f(x) = \log x$ on $[1, e^2]$.

$f(1) = 0, f(e^2) = 2$. So $f(a) \neq f(b)$ , Rolle doesn't apply. But Lagrange does: $\dfrac{1}{c} = \dfrac{2 - 0}{e^2 - 1}$, so $c = \dfrac{e^2 - 1}{2}$.

Example 5. Show that the equation $x^5 + 5x - 7 = 0$ has exactly one real root.

$f(x) = x^5 + 5x - 7$. $f'(x) = 5x^4 + 5 \ge 5 > 0$, so $f$ is strictly increasing. Strictly monotonic continuous function on $\mathbb{R}$ has exactly one root (since $f(\pm\infty) = \pm\infty$).

Example 6. Find a point $c$ on the curve $y = x^2$ over $[1, 3]$ where the tangent is parallel to the secant.

Slope of secant: $(9 - 1)/(3 - 1) = 4$. MVT: $2c = 4$, so $c = 2$. $\checkmark$

Try it yourself

1. Verify Rolle for $f(x) = x^2 - 1$ on $[-1, 1]$.
2. Apply Lagrange to $f(x) = e^x$ on $[0, 1]$ and find $c$.
3. Apply Lagrange to $f(x) = \tan x$ on $[0, \pi/4]$ and find $c$.
4. Show Rolle fails for $f(x) = |x|$ on $[-1, 1]$.
5. Use MVT to prove $|\cos a - \cos b| \le |a - b|$.
6. Show $f(x) = x^3 + x$ is strictly increasing on $\mathbb{R}$.
7. If $f'(x) = 0$ on $(a, b)$, prove $f$ is constant there.
8. Use Rolle to show that between any two roots of $\sin x$ there is a root of $\cos x$.
9. Verify Rolle for $f(x) = (x - 1)(x - 2)(x - 3)$ on $[1, 3]$.
10. Apply MVT to $f(x) = \ln x$ on $[1, 2]$ to estimate $\ln 2$.
11. Show the equation $x^4 + 4x + c = 0$ has at most two real roots.
12. If $f(0) = -3$ and $f'(x) \le 5$ for all $x$, what is the largest possible value of $f(2)$?
13. Verify Rolle for $f(x) = \cos x - 1$ on $[0, 2\pi]$.
14. Show that between any two zeros of a polynomial, the derivative has a zero (Rolle plus algebra).

Pitfalls / Tricks

• Always check all the hypotheses (continuity, differentiability, end-value match) before applying Rolle.
• MVT gives existence, not uniqueness. There may be many valid $c$.
• The most common application is to estimate or bound function differences using $f'$.
• Rolle is a special case of MVT (when $f(a) = f(b)$, the slope of the chord is $0$).
• For inequalities like $|\sin a - \sin b| \le |a - b|$, MVT plus $|f'| \le 1$ gives a one-line proof.

This concludes Chapter 5. The application of these tools to monotonicity, optimisation, and graph-sketching is the subject of Chapter 6.