proof of Fermat’s Theorem (stationary
points)∗
paolini†
2013-03-21 16:15:12
Suppose that x0 is a local maximum (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
f (x0 + h) − f (x0 )
≤ 0.
h
Since the limit of this ratio as h → 0+ exists and is equal to f 0 (x0 ) we conclude
that f 0 (x0 ) ≤ 0. On the other hand for h ∈ (−δ, 0) we notice that
f (x0 + h) − f (x0 )
≥0
h
but again the limit as h → 0+ exists and is equal to f 0 (x0 ) so we also have
f 0 (x0 ) ≥ 0.
Hence we conclude that f 0 (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.
∗ hProofOfFermatsTheoremstationaryPointsi
created: h2013-03-21i by: hpaolinii version:
h34452i Privacy setting: h1i hProofi h26A06i
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