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

定理 1

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists $\xi \in (a,b)$ with $f'(\xi) = 0$.

几何解释

If a smooth curve starts and ends at the same height, some interior point must have a horizontal tangent.

证明

1. By the Extreme Value Theorem, $f$ attains a max/min on $[a,b]$.
2. If an extreme point lies inside $(a,b)$, the derivative there is $0$.
3. If both extrema are at endpoints, $f$ is constant and any interior point works.

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)

定理 2

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $\xi \in (a,b)$ such that

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

几何解释

At some point, the tangent slope equals the slope of the secant line joining $(a,f(a))$ and $(b,f(b))$.

证明

1. Define $g(x) = f(x) - f(a) - \frac{f(b) - f(a)}{b - a}(x-a)$.
2. Then $g(a) = g(b) = 0$.
3. By Rolle, $\exists \xi$ with $g'(\xi) = 0$.
4. Hence $f'(\xi) = \dfrac{f(b) - f(a)}{b-a}$.

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

定理 3

If $f,g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x) \ne 0$, then $\exists \xi \in (a,b)$ such that

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

几何解释

For two smooth curves sharing parameter $x$, some pair of points has tangent-slope ratio equal to the secant-slope ratio.

证明

1. Let $h(x) = f(x) - f(a) - \dfrac{f(b)-f(a)}{g(b)-g(a)}(g(x)-g(a))$.
2. $h(a) = h(b) = 0$.
3. By Rolle, $h'(\xi) = 0$.
4. Rearranging yields the conclusion.

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
   $f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k + \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}.$

---

练习题

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 $f'(\xi)=0$.

参考答案

Rolle’s theorem applies directly because all hypotheses are satisfied, so $f'(\xi)=0$ for some $\xi\in(a,b)$.

练习 2

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

参考答案

Apply MVT to $f(t)=\cos t$ on $[x,y]$: $-\sin \xi = \dfrac{\cos y - \cos x}{y - x}$. Since $|\sin \xi|\le 1$, the inequality follows.

练习 3

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

参考答案

$\dfrac{2\xi}{3\xi^2} = \dfrac{3}{7}$ ⇒ $\xi = \dfrac{14}{9}$.

---

总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\xi$	希腊字母	Xi（克西）	中值定理中存在性的那个点

中英对照

中文术语	英文术语	音标	说明
微分中值定理	mean value theorem	/miːn ˈvæljuː ˈθɪərəm/	平均变化率等于某点瞬时变化率
罗尔定理	Rolle’s theorem	/rəʊlz ˈθɪərəm/	端点相等时存在水平切线
拉格朗日中值定理	Lagrange mean value theorem	/ləˈɡrɑːndʒ ˌmiːn ˈvæljuː ˈθɪərəm/	割线斜率等于某点切线斜率
柯西中值定理	Cauchy mean value theorem	/ˈkəʊʃi ˌmiːn ˈvæljuː ˈθɪərəm/	两函数导数比等于函数增量比
割线	secant line	/ˈsiːkənt laɪn/	连接区间端点的直线
单调性	monotonicity	/ˌmɒnəʊtəˈnɪsɪti/	函数增减趋势
凸函数	convex function	/ˈkɒnvɛks ˈfʌŋkʃən/	二阶导数大于零的函数

PreviousHistory of the Derivative NextL'Hôpital's Rule