91Ó°ÊÓ

• Rolle's theorem and the mean value theorem.
• Using critical points to calculate local extrema.

To study the mean value theorem of this section it is better to learn the definition of Local extrema, critical point, Rolle's theorem Therefore,

• Definition: Local extrema

1. f has local maximum at x=c if $f\left(c\right)â‰\yen f\left(x\right)$for all nearby values of x=c.
2. f has local minimum at x=c if $f\left(c\right)≤f\left(x\right)$for all nearby values of x=c.

• Rolle's theorem

if f is continuous on [a,b] and differentiable on (a,b) , and if $f\left(a\right)=f\left(b\right)=0$, then there exist at least one value $c∈(a,b)$for which $f^{\text{'}}\left(c\right)=0$.

• Mean value Theorem

If f is continuous on [a,b] and differentiable on (a,b) , and if $f\left(a\right)=f\left(b\right)=0$, then there exist at least one value $c∈(a,b)$of such that

$f^{\mathrm{′}}\left(c\right)=\frac{f\left(b\right)−f\left(a\right)}{b−a}$

• local extrema can be found using the first derivative of the given function such that $f^{\text{'}}\left(x\right)\_0$, Every local extremum is a critical point , though every critical point is not a local extremum, In that case inflection points can be found.

Given upper definitions are important for given Theorems.