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Home  Continuous Functions Derivatives and Curve Sketching Theorem 3: Same Sign Within an Interval |
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Theorem 3: Same Sign Within an Interval
Theorem 1 tells what happens when f' always has the same sign on an open interval I, while Theorem 2 does the same thing for f". To use these results we need another theorem that tells us that certain functions always have the same sign on I.
THEOREM 3 Suppose g is continuous on I, and g(x) ≠ 0 for all x in I. (i) If g(c) > 0 for 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. The two cases are shown in Figure 3.7.6. We give the proof in the next section.
Figure 3.7.6 Let us show with some simple examples how we can use the first and second derivatives in sketching curves. The three theorems above and the tests for minima and maxima are all helpful.
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Last Update: 2006-11-05