Question

What is the use of the mean value theorem?

Answer 1

Mean-value theorem:

If a function $f\left(x\right)$ is continuous in $\left[a,b\right]$ and differentiable in $\left(a,b\right)$, then there exists a number $c$ in $\left(a,b\right)$ such that,

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

The mean value theorem is used in:

• Leibniz’s rule
• The symmetry of second derivatives
• L’Hospital’s rule