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Concavity and Second Derivative Test

Concavity and Second Derivative Test. Lesson 4.4. . Concavity. Concave UP Concave DOWN Inflection point: Where concavity changes. At inflection point slope reaches maximum positive value. After inflection point, slope becomes less positive. Slope starts negative.

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Concavity and Second Derivative Test

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  1. Concavity and Second Derivative Test Lesson 4.4

  2. Concavity • Concave UP • Concave DOWN • Inflection point:Where concavitychanges

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

  4. Graph of the slope 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

  5. 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

  6. 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 isconcave down View Geogebra Demo

  7. 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

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

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

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

  11. 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

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

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