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f ’’(x) > 0 for all x in I

Study the concavity of a function using the second derivative test. Determine if the curve lies above or below the tangents, and find the inflection points.

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f ’’(x) > 0 for all x in I

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  1. Sec 4.3: Concavity and the Second Derivative Test the curve lies above the tangents the curve lies below the tangents Concavity Test Concavity Test 1 2 f ’’(x) > 0 for all x in I f ’’(x) < 0 for all x in I f(x) concave Down f(x) concave Up

  2. Sec 4.3: Concavity and the Second Derivative Test Concavity 1 f ’’(x) > 0 for all x in I f(x) concave Up 2 f(x) concave Down f ’’(x) < 0 for all x in I

  3. Sec 4.3: Concavity and the Second Derivative Test Example: Study the concavity of the function

  4. Sec 4.3: Concavity and the Second Derivative Test F092

  5. Sec 4.3: Concavity and the Second Derivative Test Inflection point: 1 _ + _ 2 +

  6. Sec 4.3: Concavity and the Second Derivative Test Find all inflection points

  7. Sec 4.3: Concavity and the Second Derivative Test Example: Find all inflection points of

  8. Sec 4.3: Concavity and the Second Derivative Test F091

  9. Sec 4.3: Concavity and the Second Derivative Test

  10. Sec 4.3: Concavity and the Second Derivative Test second Derivative Test: 1 1 2 2 second Derivative Test: 1 1 2 2 second Derivative Test: 1 1 the test fails. The function ƒ may have a local maximum, a local minimum, or neither. 2 2

  11. Sec 4.3: Concavity and the Second Derivative Test F121

  12. Sec 4.3: Concavity and the Second Derivative Test F081

  13. Sec 4.3: Concavity and the Second Derivative Test

  14. Sec 4.3: Concavity and the Second Derivative Test

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