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Understanding Concavity and Inflection Points in Calculus

Learn how concavity and inflection points impact the shape of a graph using the second derivative test in calculus. Discover methods to find local maxima and minima with examples. Explore critical points with the first and second derivative tests.

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Understanding Concavity and Inflection Points in Calculus

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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 Find all inflection points

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

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

  10. Sec 4.3: Concavity and the Second Derivative Test

  11. Sec 4.3: Concavity and the Second Derivative Test

  12. Sec 4.3:HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Two methods to find local max and local min First Derivative Test: Second Derivative Test:

  13. 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

  14. Sec 4.3: Concavity and the Second Derivative Test F121 Critical points: Local max Local min

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

  16. Sec 4.3: Concavity and the Second Derivative Test F132

  17. Sec 4.3: Concavity and the Second Derivative Test F141

  18. Sec 4.3: Concavity and the Second Derivative Test

  19. Sec 4.3: Concavity and the Second Derivative Test How to find local max and local min Find all critical points 1st deriv test 1stderiv test 2ed deriv test No test Graph or others

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