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.