TheoremProved

Mean value theorem (Lagrange)

Statement

If $f:[a,b]\to\mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then there exists $c\in(a,b)$ such that $f'(c) = \dfrac{f(b)-f(a)}{b-a}$ .

Why is it true?

On a car trip, at some instant your speedometer reading (instantaneous speed) must equal your average speed for the whole trip — you can't be strictly faster than average the entire way, nor strictly slower the entire way.

Proof sketch

Define the auxiliary function $g(x) = f(x) - f(a) - \dfrac{f(b)-f(a)}{b-a}(x-a)$ , which measures the vertical gap between $f$ and the secant line through $(a,f(a))$ and $(b,f(b))$ . Then $g$ is continuous on $[a,b]$ , differentiable on $(a,b)$ , and $g(a)=g(b)=0$ , so Rolle's theorem gives $c\in(a,b)$ with $g'(c)=0$ , i.e. $f'(c) = \dfrac{f(b)-f(a)}{b-a}$ .

Stated by

stated 1797Joseph-Louis Lagrange

Topics that use this theorem

Applications of the Derivative Definite Integral

Related theorems

Rolle's theorem Fundamental theorem of calculus L'Hôpital's rule

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Joseph-Louis Lagrange (1797). Théorie des fonctions analytiques
James Stewart (2015). Calculus