1. Section 13.1 – 13.2 Increasing/Decreasing Functions and Relative Extrema

2. Facts If f ’( x ) > 0 on an interval ( a,b ), then f (x) is increasing on ( a,b ). If f ’( x ) < 0 on an interval ( a,b ), then f (x) is decreasing on ( a,b ). A number c for which f ’( c ) = 0 or f ’( c ) = undefined is called the critical number (critical value). Example: Find the intervals where the function is increasing/decreasing Definition:

3. Definition  A function f has a relative maximum (or local max) at c if f ( c ) > f ( x ) for all x near c.  A function f has a relative minimum (or local min) at c if f ( c ) < f ( x ) for all x near c.

4. The First Derivative Test: If f ’(c) changes from + to – at c, then f has a local maximum at c. If f ’(c) changes from – to + at c, then f has a local minimum at c.  No sign change at c means no local extremum (maximum or minimum)

5. How to find local max/min and interval of increasing/decreasing: 1) Find all critical values by solving f ’(x) = 0 or f ’(x) = undefined 2) Put all critical values on the number line and use test values to determine the sign of the derivative for each interval. 3) Determine the interval of increasing/decreasing based on the sign of derivative.

6. Examples Find the intervals of increase/decrease and all local extrema.

7. 7 Examples A small company manufactures and sells bicycles. The production manager has determined that the cost and demand functions for q ( q > 0) bicycles per week are where p is the price per bicycle. a)Find the (weekly) revenue function. b)Find the maximum weekly revenue. c)Find the maximum weekly profit. d)Find the price the company should charge to realize maximum profit.