Mean Value Theorem

A continuous, differentiable function's instantaneous rate of change matches its average rate of change somewhere on the interval.

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then there exists at least one $c \in (a,b)$ with $f'(c) = \dfrac{f(b)-f(a)}{b-a}$ — the tangent line at $c$ is parallel to the secant line through the endpoints.

Examples

For $f(x)=x^2$ on $[0,2]$ : average rate of change is $2$ , and $f''(c)=2c=2$ gives $c=1$ .