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Proof of the Critical Point Theorem;

Assume that neither (i) nor (ii) holds; that is, assume that c is not an endpoint of I and f'(c) exists. We must show that (iii) is true; i.e., f'(c) = 0. We give the proof for the case that f has a maximum at c. Let x = c, and let Δx > 0 be infinitesimal. Then

f(c + Δx) ≤ f(c), f(c - Δx) ≤ f(c).

(See Figure 3.5.14.) Therefore

03_continuous_functions-165.gif

03_continuous_functions-166.gif

Figure 3.5.14

Proof of the Critical Point Theorem Taking standard parts,

03_continuous_functions-167.gif ,

and also,

03_continuous_functions-168.gif

Therefore f'(c) = 0.


Last Update: 2010-11-25