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Section 13.1 – 13.2 Increasing/Decreasing Functions and Relative Extrema

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

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Section 13.1 – 13.2 Increasing/Decreasing Functions and Relative Extrema

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  1. Section 13.1 – 13.2Increasing/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: • Find all critical values by solving f’(x) = 0 or f ’(x) = undefined • Put all critical values on the number line and use test values to determine the sign of the derivative for each interval. • 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. 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. Find the (weekly) revenue function. Find the maximum weekly revenue. Find the maximum weekly profit. Find the price the company should charge to realize maximum profit.

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