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3.4 Concavity and the Second Derivative Test The second derivative is useful for determining: • the concavity of a function, • the location of possible points of inflection, • or whether a critical number on the graph is indeed a relative maximum or relative minimum. Test for Concavity Let f be a function whose second derivative exists on an open interval I. 1. If f ’’(x) > 0 for all x in I, then the graph is concave up in I. 2. If f ’’(x) < 0 for all x in I, then the graph is concave down in I. To determine the concavity for a function, locate the x-values for which f ’’(x) = 0 or f ’’(x) is undefined. Test the intervals between these values. Points of Inflection A possible point of inflection on a graph can be determined by finding the x values for which the f ’’(x) = 0 or is undefined. Test for Relative Maximum or Minimum Let f be a function such that f ’(c) = 0 and the second derivative exists on an open interval containing c. 1. If f ’’(c) > 0, then f(c) is a relative minimum (function is concave up at that point). 2. If f ’’(c) < 0, then f(c) is a relative maximum (function is concave down at that point). 3. If f ’’(c) = 0, then the test fails, you must use the First Derivative Test. Example 1: Determine the points of inflection and discuss the concavity of the graph: f ( x ) = 4 x 3 ! 12 x 11/2006 Example 2: Use the second derivative to find the relative extrema and points of inflection, if appropriate, for the function: f ( x ) = x 3 ! 12 x ! 5 Example 3: Use the second derivative to find the relative extrema and points of inflection, if appropriate, for x the function: f ( x ) = 2 x +1 11/2006