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Information on the Second Derivative Test for determining the concavity and inflection points of functions. It includes examples of finding points of inflection and intervals of concavity up and down for various functions.
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f(x) is concave up whenever f'(x) is increasing. f(x) is concave down whenever f'(x) is decreasing. The test for concavity: If f"(x) is positive for all x in (a,b) then f(x) is concave up in (a,b). If f"(x) is negative for all x in (a,b) then f(x) is concave down in (a,b). A point of inflection occurs where the concavity changes. If (c, f(c)) is a point of inflection, then both #1 and #2 are true:
The 2nd Derivative Test In order to use this test, f'(c) must equal 0. If f"(c) > 0, then f(c) is a MINIMUM. If f"(c) (^) < 0, then f(c) is a MAXIMUM. If f"(c) = 0, then the TEST FAILS. (You'll need to go back and use the 1st Derivative Test.) Find all relative extrema. Use the 2nd Derivative Test whenever possible: