The integral mean value theorem

Having defined the integral mean, the theorem states that it is actually attained as the value of the function at at least one point of the interval.

Theorem — Integral mean value theorem

If $f$ is continuous on $[a,b]$, then there exists at least one point $c\in[a,b]$ such that: $f(c) = \bar f_{ab} = \frac{1}{b-a}\int_a^b f(x)\,dx.$ In other words: the function takes at at least one point the value of its integral mean.

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Topics: Calculus theorems
Concepts: Integral mean · Integral mean value theorem
Skills: Integrating