Concavity and the Second Derivative

1. Concavity and the Second Derivative mysite.oswego308.org/.../3.4- ...

2. Relative max. f (0) = 1 Relative min. f (4) = -31 The First Derivative Find the intervals of increase and decrease of + 0 - 0 + 0 4 inc: (-∞,0)(4,∞) dec: (0,4) http://math.colstate.edu/calmada/courses/math1125/ch5%20sect%201-2-3-4.ppt#338,1,Chapter 5 Applications of the Derivative Sections 5.1, 5.2, 5.3, and 5.4

3. Concavity of a Function As you look at the graph of a function …  … if the function CURVES UP, like a cup, we say the function is _______________.   …if the function CURVES DOWN, like a frown, we say the function is _______________. CONCAVE UP CONCAVE DOWN

4. What do their eyes mean??? + + – –

5. is positive is positive is negative is negative is zero Second derivative: Curve is concave up. Curve is concave down. Possible inflection point (where concavity changes). + + – –

6. Points of Inflection

7. Example: inc: (-∞,0)(2,∞) dec: (0,2) positive negative positive

8. Example: up: (1,∞) down: (-∞,1)

9. Example: Determine the points of inflection and discuss the concavity of the graph of (x^4-4x^3)/15 Points of inflection

10. Homework: • Page 195 • #5-10, 21, 22