## Lecture Notes: Wednesday, October 7, 1998

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

### Exam # 2 tonight (NEGB AUD at 6:20 p.m.)

### 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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