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

Rolle's theorem

Rolle's theorem

Let \( a < b \) be real numbers, and let \( g : [a, b] \to \mathbb{R} \) be a function such that:

  1. \( g \) is continuous on \( [a, b] \) and differentiable on \( (a, b) \).
  2. \( g(a) = g(b) \).
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 \( f : X \to \mathbb{R} \) be a function and let \( x_0 \in X \). We say that \( f \) obtains a local maxima at \( x_0 \) if there exists some \( \delta > 0 \) such that the restricted function \( f |_{X \cap (x_0 - \delta, x_0 + \delta)} \) 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 \( f : (a, b) \to \mathbb{R} \) 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