# 4.4 Notes - Example For the following examples find the...

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Calculus I Section 4.4 – Concavity and the 2 nd Derivative Test Test for Concavity Let f be a function whose second derivative on an open interval I . The graph of f is concave upward if ______________________________________. The graph of f is concave downward if ___________________________________ . Points of Inflection An inflection point is a point on the graph where the concavity changes. If )) ( , ( c f c is a point of inflection on the graph of f , then either 0 ) ( c f or ) ( c f is undefined. Caution: The inverse of the above theorem is not true. Just because 0 ) ( c f or ) ( c f is undefined does not mean you have an inflection point!! The concavity must change at c for it to be an inflection point.

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Unformatted text preview: Example: For the following examples find the intervals where the function is increasing/decreasing, relative extrema, points of inflection, and determine the concavity. 5 12 3 2 ) ( 2 3 x x x x f 1 ) ( x x x f Second Derivative Test Let f be a function such that ) ( c f and the second derivative of f exists on an open interval containing c . If ) ( c f > 0, then ) ( c f is a relative _________________________. If ) ( c f < 0, then ) ( c f is a relative _________________________. If ) ( c f = 0, the test fails. Use the First Derivative Test. 2 2 4 2 8 1 ) ( x x x g x x y ln x x x f 2 arcsin ) (...
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4.4 Notes - Example For the following examples find the...

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