Lecture Notes: Wednesday, October 7, 1998

Fifth Homework Assignment - for discussion at recitations on Thursday October 8, Tuesday October 13, and Thursday October 15..

Chapter 4

• Section 2 (Relative Maxima and Minima)
• FACT: If a differentiable function f(x) has a relative max or min at x=a, then f'(a) = 0.
• Converse is false. Ex: f(x) = x^3 at x=0. Note sign of f' on either side of this critical point.
• Section 3 (Concavity and Points of Inflection)
• Concavity:
• Concave up -- f' increasing -- f" positive
• Concave down -- f' decreasing -- f" negative
• Inflection point: means change of concavity.
• f" goes from negative to zero to positive or vice versa.
• Application to extrema:
• f differentiable. c inside domain of f. f'(c) = 0, then:
• f"(c) > 0 -> c is local min
• f"(c) < 0 -> c is local max

Examples:

1. Where is x^3 - 12*x^2-15*x+7 increasing, decreasing, concave up, concave down? Where are the inflection points, if any? What are the local max and min, if any? Use MAPLE to help.

2. In MAPLE, plot cos(1.1*x+.5), its derivative, and second derivative. From plot, determine which is which.

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