Lagrange's Mean Value Theorem

Theorem 7.3 (Lagrange's Mean Value Theorem). Let $f(x)$ be continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$ (where $f(a),f(b)$ are not necessarily equal). Then there exists at least one point $c\in(a,b)$ such that

Physical meaning. $\dfrac{f(b)-f(a)}{b-a}$ is the average rate of change of $f$ over $[a,b]$, while $f'(c)$ is an instantaneous rate of change; LMVT says these two must coincide at some interior instant.

Geometric meaning. The secant joining $\big(a,f(a)\big)$ and $\big(b,f(b)\big)$ has exactly the same slope as the tangent at some interior point $c$ — the tangent at $c$ is parallel to that secant. (For instance, a car accelerating from rest that covers 200 m in 8 s has an average velocity of $25$ m/s; the Mean Value Theorem guarantees that at some instant during the trip the speedometer reads exactly $25$ m/s $=90$ km/h.)

Theorem 7.4 (a monotonicity consequence, restated in §7.6.1's language). If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f'(x)>0$ for every $x\in(a,b)$, then for any $x_1<x_2$ in $[a,b]$, $f(x_1)<f(x_2)$ — proved directly from LMVT: pick $c$ between $x_1,x_2$ with $f(x_2)-f(x_1)=f'(c)(x_2-x_1)$; since $f'(c)>0$ and $x_2-x_1>0$, the right side is positive.

Three further consequences of LMVT, used throughout the rest of the chapter: …