Concavity and Second Derivative Test

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Presentation transcript:

Concavity and Second Derivative Test Lesson 4.4

Concavity Concave UP Concave DOWN Inflection point: Where concavity changes 

Inflection Point Consider the slope as curve changes through concave up to concave down At inflection point slope reaches maximum positive value After inflection point, slope becomes less positive  Slope starts negative Graph of the slope Slope becomes positive, then more positive Becomes less negative Slope becomes (horizontal) zero

Inflection Point What could you say about the slope function when the original function has an inflection point Slope function has a maximum (or minimum Thus second derivative = 0 Graph of the slope

Second Derivative This is really the rate of change of the slope When the original function has a relative minimum Slope is increasing (left to right) and goes through zero Second derivative is positive Original function is concave up

Second Derivative When the original function has a relative maximum The slope is decreasing (left to right) and goes through zero The second derivative is negative The original function is concave down View Geogebra Demo

Not an inflection point Second Derivative If the second derivative f ’’(x) = 0 The slope is neither increasing nor decreasing If f ’’(x) = 0 at the same place f ’(x) = 0 The 2nd derivative test fails You cannot tell what the function is doing Not an inflection point

Example Consider Determine f ‘(x) and f ’’(x) and when they are zero

Example f ’(x) = 0, f’’(x) > 0, this is concave up, a relative minimum f ‘(x) f ‘(x) = 0, f ‘’(x) < 0 this is concave down, a maximum f(x)  f ‘’(x) = 0 this is an inflection point f ‘’(x)

Example Try f ’(x) = ? f ’’(x) = ? Where are relative max, min, inflection point?

Algorithm for Curve Sketching Determine critical points Places where f ‘(x) = 0 Plot these points on f(x) Use second derivative f’’(x) = 0 Determine concavity, inflection points Use x = 0 (y intercept) Find f(x) = 0 (x intercepts) Sketch

Assignment Lesson 4.4 Page 235 Exercises 5, 12, 19, 26, 33, 40, 47, 54, 61, 68, 73, 83, 91, 95, 96