Math and science::Analysis::Tao::10: Differentiation of functions

Rolle's theorem

Let $a < b$ be real numbers, and let $g : [a, b] \to R$ be a function such that:

$g$ is continuous on $[a, b]$ and differentiable on $(a, b)$ .
$g (a) = g (b)$ .

Then there exists an

x \in (a, b)

such that

g^{'} (x) = 0

A corollary of Rolle's theorem is the mean value theorem.

I think the following concepts are used when proving Rolle's theorem.

Local maxima and minima

Let $f : X \to R$ be a function and let $x_{0} \in X$ . We say that $f$ obtains a local maxima at $x_{0}$ if there exists some $δ > 0$ such that the restricted function $f |_{X \cap (x_{0} - δ, x_{0} + δ)}$ of $f$ obtains a maximum at $x_{0}$ .

Local minima are defined similarly.

Local extrema are stationary

Let $a < b$ be real numbers, let $x_{0} \in (a, b)$ , and let $f : (a, b) \to R$ be a function which is differentiable at $x_{0}$ . If $f$ attains either a local maximum or local minimum at $x_{0}$ , then $f^{'} (x_{0}) = 0$ .

Source

p260

15.04.2020