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Concavity and the 2nd Derivative Test

# Concavity and the 2nd Derivative Test - the graph of f then...

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Concavity and the 2 nd Derivative Test Section 3.4

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Defn. of Concavity Let f be differentiable on an open interval I . The graph of f is concave upward on I if f ’ is increasing on the interval and concave downward on I if f ’ is decreasing on the interval concave up concave down f ’ is increasing f ’ is decreasing
Test for Concavity Let f be a function whose derivative exists on an open interval I . 1. If f ’’(x) > 0 for all x in I , then the graph of f is concave upward in I . 2. If f ’’(x) < 0 for all x in I , then the graph of f is concave downward in I .

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Points of Inflection If the tangent line exists at a point which concavity changes, that point is an inflection point . point of inflection
If (c, f(c) ) is a point of inflection of

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Unformatted text preview: the graph of f , then either f ’’ (c) = 0 or f ’’ is undefined at x = c • To locate possible points of inflection, determine the values of x for which f ’’(x) = 0 or is undefined. 2 nd Derivative Test Let f be a function such that f ’’(c) = 0 and the 2 nd derivative of f exists on an open interval containing c . 1. If f ’’(c) > 0, then f(c) is a relative minimum. 2. If f ’’(c) < 0, then f(c) is a relative maximum. If f ’’(c) = 0, the test fails. In such cases, you can use the 1 st Derivative Test. Joke Time What do you do if you are outside during a thunderstorm? co-incide Why do lumberjacks make good musicians? because of their natural log-a-rithms...
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Concavity and the 2nd Derivative Test - the graph of f then...

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