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Home Continuous Functions Derivatives and Curve Sketching Derivatives and Curve Sketching - Part_10
 
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Derivatives and Curve Sketching - Part_10

In general a curve y = f(x) may go up and down several times. To sketch it we need to determine the intervals on which it is increasing or decreasing, and concave upward or downward. Here are some things which may happen at the endpoints of these intervals.

DEFINITION

Let c be an interior point of I.

  • f has a local maximum at c if f(c) ≥ f(x) for all x in some open interval (a0, b0) containing c.
  • f has a local minimum at c if f(c) ≤ f(x) for all x in some open interval (a0, b0) containing c. (The interval (a0, b0) may be only a small subinterval of I.)
  • f has a point of inflection at c if f changes from one direction of concavity to the other at c.

These definitions are illustrated in Figure 3.7.9.

03_continuous_functions-251.gif

Figure 3.7.9
Home Continuous Functions Derivatives and Curve Sketching Derivatives and Curve Sketching - Part_10

Last Update: 2006-11-05