Question

What does the Mean Value Theorem mean?

Answer 1

Mean Value Theorem:

The theorem states that if a function $f$ is continuous on the closed interval $\left[a,b\right]$ and differentiable on the open interval $\left(a,b\right)$, then there exists a point $c$ in the interval $\left(a,b\right)$ such that $f\text{'}\left(c\right)$ is equal to the function's average rate of change over $\left[a,b\right]$.

The slope of the secant is equal to the slope of the tangent of the given curve.

The average rate of change of the slope of the secant in the interval is $\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

By the Mean value theorem, there exists a $c\in \left(a,b\right)$ such that the slope of the secant line through the endpoints $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$ is equal to the slope of the tangent at $c$ i.e., $f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$