Question

What is the conclusion of the Mean Value Theorem?

Answer 1

Mean Value Theorem:

if a function $f$ is continuous on the interval $\left[a,b\right]$, and also differentiable on the interval $\left(a,b\right)$, there exists at least one point $c$ in the interval $\left(a,b\right)$, such that

$f\text{'}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Conclusion of the Mean Value Theorem:

The conclusion is that there exists a point $c$ in the interval $\left(a,b\right)$ such that the tangent at the point $\left(c,f\left(c\right)\right)$ is parallel to the line that passes through the points $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$.