Section 3.4

1. Section 3.4 1st Day Read p. 190.

2. Test for Concavity • Let f be a function whose second derivative exists on an open interval I. • 1. If f ꞌꞌ(x) > 0 for all x in I, then the graph of f is concave upward in I. • 2. If f ꞌꞌ(x) < 0 for all x in I, then the graph of f is concave downward in I. • Read the paragraph and Note after this theorem on p. 191.

3. Example 1 • Determine the open intervals on which the graph of • is concave upward or downward.

4. Then find the 2nd derivative. 1st make sure the graph is continuous.

8. Use the intervals (-∞, -1), (-1, 1), and (1, ∞) to find where the graph is concave up or concave down.

9. f is concave up on the intervals (-∞, -1); (1, ∞). • f is concave down on the interval (-1, 1). + − + 1 -1

10. Concave upward Concave upward Concave downward

11. Example 2 • Determine the open intervals on which the graph of • is concave upward or downward.

14. There are no values of x that will make the 2nd derivative equal to 0 but at x = -2 and x = 2 the function is not continuous. Therefore the intervals will be (-∞, -2), (-2, 2), and (2, ∞).

15. f is concave up on the intervals (-∞, -2); (2, ∞). • f is concave down on the interval (-2, 2). + − + 2 -2

16. Concave upward Concave upward Concave downward

17. Definition of Point of Inflection • Let f be a function that is continuous on an open interval, let c be a point in the interval. If the graph of f has a tangent line at this point (c, f(c)), then this point is a point of inflection of the graph if the concavity of f changes from upward to downward (or downward to upward) a the point. • Read the Note on p. 192 and look at the types of points of inflection.

18. Point of Inflection • If (c, f(c)) is a point of inflection of the graph of f, then either f ꞌꞌ(c) = 0 or f ꞌꞌ does not exist at • x = c.

19. Example 3 • Determine the points of inflection and discuss the concavity of the graph of

20. So the intervals are (-∞, 0), (0, 2), and (2, ∞).

21. f is concave up on the intervals (-∞, 0); (2, ∞). • f is concave down on the interval (0, 2). + − + 2 0

22. Concave upward Concave upward Concave downward The points of inflection are at (0, 0) and (2, -16). Read the at last paragraph on p. 193.

23. 2nd Day • Read the 1st paragraph on p. 194.

24. Second Derivative Test • Let f be a function such that f ꞌ(c) = 0 and the second derivative of f exists on an open interval containing c. • 1. If f ꞌꞌ(c) > 0, then f has a relative minimum at (c, f(c)). • 2. If f ꞌꞌ(c) < 0, then f has a relative maximum at (c, f(c)). • If f ꞌꞌ(c) = 0, the test fails. That is, f may have a relative minimum, relative maximum, or neither. In such cases, you can use the First Derivative Test.

25. Example • Find the relative extrema for f(x) = -3x5 + 5x3. • Then find the intervals that the graph is concave up and concave down. • Finally find the points of inflection.

26. 1. Find the critical numbers for f by finding the 1st derivative. • f ꞌ(x) = -15x4 + 15x2 • x = -1, 0, 1 • 2. Use the 2nd Derivative Test to find the relative minimum or relative maximum. • f ꞌꞌ(x) = -60x3 + 30x

27. At x = -1 there is a relative minimum. • At x = 0 we need to use the 1st Derivative Test. • At x = 1 there is a relative maximum. + − = -1 1 0

28. 3. Find the critical #s for the second derivative

29. + + − − 0

30. pp. 195-197 (4-56 mult. of 4)