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1. 3.4 Concavity and the 2nd Derivative Test

2. concave up: f’ is increasing. tangent lines are below the graph. concave down: f’ is decreasing. tangent lines are above the graph.

3. Test for Concavity • If f”(x) > 0 for all x in I, then f is concave up. • If f”(x) < 0 for all x in I, then f is concave down. 2nd Derivative Test (Use C.N.’s of 1st Derivative) • If f”(c) > 0, then f(c) is a relative min. • If f”(c) < 0, then f(c) is a relative max. • If f”(c) = 0, then the test fails. No min. or max.

4. Ex. 1 Determine where f(x) = 6(x2 + 3)-1 is increasing, decreasing, has max’s or min’s, is concave up or down, has inflection points. f’(x) = -6(x2 + 3)-2(2x) C.N. = 0 + 0 (0,2) max. 1st der. test

5. C.N.’s -1, 1 -1 1 down concave up up (-1, ) (1, ) Inflection points

6. 2nd Derivative Test Plug C.N.’s of 1st der. into 2nd derivative. C.N. from 1st der. was 0. 0 is a maximum Remember, a neg. in the 2nd der. means concave down. Therefore, the point is a maximum.

7. There are no x-intercepts (0,2) max (1, 3/2) Inf. pt. (-1, 3/2) Inf. pt.

8. Ex. 2 Determine where f(x) = x4 – 4x3 is increasing, decreasing, has max’s or min’s, is concave up or down, has inflection points. f’(x) = 4x3 – 12x2 = 4x2(x – 3) C.N.’s 0, 3 0 3 dec. dec. inc. (3, ) -27 min. 1st der. test f”(x) = 12x2 – 24x = 12x(x – 2) = 0 C.N.’s 0, 2 2nd der. test up down up 0 2 f”(0) f”(3) = 0 > 0 Inf. pt.’s (0, ) (2, ) 0 -16 (3,-27) is a min.

9. 9 -9 -16 -27

10. Given the graph of f, trace the graph of f’, and f”, on the same axes. f’ f f”