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4.2_Concavity__summer

4.2_Concavity__summer - 4.2 Concavity DEFINITION Concavity...

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4.2 Concavity DEFINITION: Concavity of a function Let the function f be differentiable on an interval (a,b). Then 1. f is concave upward on (a,b) if f’(x) is increasing on (a,b). (PALM UP) f is concave downward on (a,b) if f’(x) is decreasing on (a,b). (PALM DOWN) A point on the graph of a differentiable function at which the concavity changes is called an inflection point Look at the graph of y = f(x) Find the largest open interval where f is concave up. Find the largest open interval where f is concave down up. Find al infection points of f. THEOREM If f ”(x) > 0 for each value of x in (a,b), then f is concave upward on (a,b). If f ”(x) < 0 for each value of x in (a,b), then f is concave downward on (a,b). Procedure for determining the Intervals of Concavity when no graph is given 1. Determine the values of x for which f”(x) = 0 or f”(x) DNE 2. Identify the open intervals determined by these numbers. 3. Determine the sign of f”(x) in each of these intervals by computing f”(c) where c is a test value chosen in each interval.
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4. If f”(c) > 0, f is concave upward on (a,b)
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