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Concavity & Inflection Points

Concavity & Inflection Points. Objectives. To determine the intervals on which the graph of a function is concave up or concave down. To find the inflection points of a graph of a function. To determine where a function has extrema using the second derivative test. Concavity.

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Concavity & Inflection Points

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  1. Concavity &Inflection Points

  2. Objectives • To determine the intervals on which the graph of a function is concave up or concave down. • To find the inflection points of a graph of a function. • To determine where a function has extrema using the second derivative test

  3. Concavity • The notion of curving upward or downward is defined as the concavity of the graph of a function.

  4. Concavity curved upward or concave up

  5. Concavity curved downward or concave down

  6. Concavity curved upward or concave up

  7. Concavity • Question: Is the slope of the tangent line increasing or decreasing?

  8. Concavity What is the derivative doing?

  9. Concavity • Question: Is the slope of the tangent line increasing or decreasing? • Answer: The slope is increasing. • The derivative must be increasing.

  10. Concavity • Question: How do we determine where the derivative is increasing? • f (x) is increasing if f’ (x) > 0. • f’ (x) is increasing if f” (x) > 0. • Answer: We must find where the second derivative is positive.

  11. Concavity • The concavity of a graph can be determined by using the second derivative. • If the second derivative of a function is positive on a given interval, then the graph of the function is concave up on that interval. • If the second derivative of a function is negative on a given interval, then the graph of the function is concave down on that interval.

  12. The Second Derivative • f (x) is concave up if f” (x) > 0. • f (x) is concave down if f” (x) < 0.

  13. Concavity Concave down Here the concavity changes. Concave up This is called an inflection point (or point of inflection).

  14. Concavity Concave up Inflection point Concave down

  15. Inflection Points • Points where the graph changes concavity are called inflection points. • The second derivative will either equal zero or be undefined at an inflection point.

  16. 1) Use the graph to find the intervals where the graph is concave up, and those where it is concave down. Also find the inflection Points.

  17. 2) Use the graph to find the intervals where the graph is concave up, and those where it is concave down. Also find the inflection Points.

  18. Inflection Point 3) Find the points of inflection. Points of Inflection

  19. Concavity 4) Find the intervals on which the function is concave up or concave down:

  20. Concavity

  21. Concavity 5) Find the intervals on which the function is concave up or concave down:

  22. Inflection Point

  23. Concavity 0

  24. Concavity

  25. Concavity • Find the intervals on which the function is concave up or concave down:

  26. Inflection Point

  27. Concavity UND.

  28. Concavity

  29. Second Derivative Test 7) Find all the relative extrema of each function using the second derivative test: a) Concave down at x=5 so it must be a maximum at 5! Only possible place for extrema Answer: Rel Max at (5, 0)

  30. 7) Find all the relative extrema of each function using the second derivative test: b) Concave up at x=0 so it must be a minimum! Concave down at x=-10/9 so it must be a maximum! Only possible places for extrema Answer: Rel Max at (-10/9, 14/243) Rel Min at (0, -2)

  31. Conclusion • The second derivative can be used to determine where the graph of a function is concave up or concave down and to find inflection points. • Knowing the critical points, local extreme values, increasing and decreasing regions, the concavity, and the inflection points of a function enables you to sketch accurate graphs of that function.

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