Mean Value Theorems for Derivatives

Mean value theorems connect an average rate of change on an interval with an instantaneous rate of change at some interior point. They underpin many proofs and estimates in calculus.

Rolle’s Theorem

Example

Show that $x^3 - 3x + 1 = 0$ has at most one real root in $(0,1)$.

• Assume two roots $x_1 \ne x_2$ in $(0,1)$.
• Then $f(x_1) = f(x_2) = 0$. Rolle gives $\xi$ with $f'(\xi) = 0$.
• But $f'(x) = 3(x^2 - 1) < 0$ on $(0,1)$, contradiction.

Lagrange Mean Value Theorem (MVT)

Examples

1. Prove $|\sin x - \sin y| \le |x - y|$.
   Apply MVT to $f(t) = \sin t$ on $[x,y]$, so $\cos \xi = \dfrac{\sin y - \sin x}{y - x}$ with $|\cos \xi| \le 1$.

2. For $f(x) = x^3$ on $[1,2]$, MVT gives $3\xi^2 = 7$, so $\xi = \sqrt{\tfrac{7}{3}}$.

Cauchy Mean Value Theorem

Example

Take $f(x) = x^2$, $g(x) = x^3$ on $[1,2]$. Cauchy MVT gives $\dfrac{2\xi}{3\xi^2} = \dfrac{3}{7}$, so $\xi = \dfrac{14}{9}$.

Relations among the theorems

• Rolle is the base case.
• Lagrange MVT generalizes Rolle by dropping $f(a)=f(b)$.
• Cauchy MVT generalizes Lagrange to two functions; letting $g(x)=x$ recovers Lagrange.

Applications

1. Inequalities: $e^x > 1 + x$ for $x>0$ (apply MVT to $e^x$ on $[0,x]$).
2. Root existence: sign change plus continuity ensures a root (Intermediate Value), and MVT refines counts.
3. Monotonicity: if $f'(x)>0$ on an interval, $f$ is strictly increasing there.
4. Convexity: $f''(x)>0$ implies $f$ is convex (and secants lie above the graph).

Common pitfalls

• Forgetting to check continuity on the closed interval.
• Forgetting differentiability on the open interval.
• Applying Cauchy MVT without ensuring $g'(x)\ne 0$.
• Using an interval where $f(a)=f(b)$ fails for Rolle.

Extensions

1. Integral mean value theorem: $\int_a^b f(x)\,dx = f(\xi)(b-a)$ for continuous $f$.
2. Taylor’s theorem remainder: $\exists \xi$ between $x$ and $x_0$ such that
   $f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k + \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}.$

---

Exercises

练习 1

Assume $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$. Show that $\exists \xi \in (a,b)$ with $f'(\xi)=0$.

练习 2

Prove $|\cos x - \cos y| \le |x - y|$.

练习 3

Verify Cauchy MVT for $f(x) = x^2$, $g(x) = x^3$ on $[1,2]$.

---

Summary

本文出现的符号

中英对照

PreviousLeibniz and the Rise of DifferentialsLeibniz’s geometric route to the derivative and the differential notation we still use.

NextL'Hôpital's Rule