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Proof of Theorem 3 on Curve Sketching
The Intermediate Value Theorem can be used to prove Theorem 3 of Section 3.7 on curve sketching: Suppose g is a continuous function on an interval I, and g(x) ≠ 0 for all x in I. (i) If g(c) > 0 or at least one c in I, then g(x) > 0 for all x in I. (ii) If g(c) < 0 for at least one c in I, then g(x) < 0 for all x in I. PROOF (i) Let g(c) > 0 for some c in I. If g(x1) < 0 for some other point x1 in I, then by the Intermediate Value Theorem there is a point x2 between c and x1 such that g(x2) = 0, contrary to hypothesis (Figure 3.8.9). Therefore we conclude that g(x) > 0 for all x in I.
Figure 3.8.9
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Last Update: 2006-11-05