3.4 Concavity and the second derivative test.notebook October 01, 2014 Section: 3.4 Concavity and the second derivative test Objective: • Determine intervals on which a function is concave upward or concave downward • Find any points of inflection of the graph of a function • Apply the second derivative test to find relative extrema of a function Homework: Pg 194 # 354 x 3 55, 59 Copy into notes. or page of things to memorize 1 3.4 Concavity and the second derivative test.notebook October 01, 2014 Concavity: Let f be differentiable on an open interval I. The graph 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. Second derivative test (informal): • If f " is positive on an interval, then the function is concave upward. • If f " is negative on an interval, then the function is concave downward. Point of inflection (Informal): When a graph changes concavity there is a point of inflection. • If (c,f(c)) is a point of inflection on the graph of f , then either f "(c) = 0 or DNE at x = c 2 3.4 Concavity and the second derivative test.notebook October 01, 2014 3 3.4 Concavity and the second derivative test.notebook October 01, 2014 4 3.4 Concavity and the second derivative test.notebook October 01, 2014 f " is positive when f ' is increasing => concave up f " is negative when f ' is decreasing => concave down 5 3.4 Concavity and the second derivative test.notebook October 01, 2014 is increasing If is positive If is negative If is = 0 or DNE If is positive If is negative if is = 0 or DNE Possible point of inflection If is increasing Is positive If is decreasing Is negative is decreasing Critical number (possible min or max Concave up Concave down 6 3.4 Concavity and the second derivative test.notebook October 01, 2014 7 3.4 Concavity and the second derivative test.notebook October 01, 2014 8 3.4 Concavity and the second derivative test.notebook October 01, 2014 9 3.4 Concavity and the second derivative test.notebook October 01, 2014 10 3.4 Concavity and the second derivative test.notebook October 01, 2014 11 3.4 Concavity and the second derivative test.notebook October 01, 2014 12 3.4 Concavity and the second derivative test.notebook October 01, 2014 13
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