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Unformatted text preview: 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. • 4. If f”(c) > 0, f is concave upward on (a,b)4....
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This note was uploaded on 10/07/2011 for the course MATH 1431 taught by Professor Vaughn during the Spring '08 term at LSU.
- Spring '08