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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 Figure 3.5.14 Proof of the Critical Point Theorem Taking standard parts, , and also,
Therefore f'(c) = 0.
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Home Continuous Functions Maxima and Minima Proof of the Critical Point Theorem; |