proof of Fermat’s Theorem (stationary points)

Suppose that x0 is a local maximumMathworldPlanetmath (a similar proof applies if x0 is a local minimum). Then there exists δ>0 such that (x0-δ,x0+δ)(a,b) and such that we have f(x0)f(x) for all x with |x-x0|<δ. Hence for h(0,δ) we notice that it holds


Since the limit of this ratio as h0+ exists and is equal to f(x0) we conclude that f(x0)0. On the other hand for h(-δ,0) we notice that


but again the limit as h0+ exists and is equal to f(x0) so we also have f(x0)0.

Hence we conclude that f(x0)=0.

To prove the second part of the statement (when x0 is equal to a or b), just notice that in such points we have only one of the two estimates written above.

Title proof of Fermat’s Theorem (stationary points)
Canonical name ProofOfFermatsTheoremstationaryPoints
Date of creation 2013-03-22 13:45:09
Last modified on 2013-03-22 13:45:09
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 5
Author paolini (1187)
Entry type Proof
Classification msc 26A06