Section 3.4 – Optimization

Absolute Maximum and Minimum of a Function

f(c) is the absolute maximum on an interval if f(c ) > f(x) for all x in interval

f(c) is the absolute minimum on an interval if f(c ) < f(x) for all x in interval

Visually - Absolute extrema are very dependent on the interval

Examine the above function for absolute extrema on various closed intervals

Procedure to find absolute maximum and minimum of f(x) on [a,b]

1) Find any points c where f ‘ (c) = 0 and c is in the interval [a,b]

2) Evaluate f(a), f(c) for any c values from 1), and f(b).

3) Find the largest (absolute max) and smallest (absolute min) of the value

in 2)

Examples

#2

#4

#8 NOTE: The interval does NOT include t = 1

If the interval under consideration is NOT Closed:

Procedure changes:

1) Find Critical numbers on interval

2) Find where f increases and decreases on the interval

3) Check with graph

#12

#16

Applications

#18 For the function

a) Find P(q), R’(q), C’(q), and sketch or same graph and find where profit is

maximized.

b) Find average cost and sketch A(q) and C’(q) on same graph.

Find where A(q) is minimized.

#32 BROADCASTING

An all-new radio station has made a survey of the listening habits of local residents between the hours of 5:00 p.m. and midnight. The survey indicates that the percentage of the local adult population that is tuned in to the station x hours after 5:00 p.m. is

a) At what time between 5:00 p.m. and midnight are the most people listening to the station? What percentage of the population is listening at this time?

b) At what time between 5:00 and midnight are the fewest people listening? What percentage of the population is listening at this time?

Efficiency Optimization

An efficiency study of the morning shift (from 8:00 a.m. to 12:00 noon) at a certain factory indicates that an average worker who arrives on the job at 8: 00 a.m. will have assembled transistor ratios t hours later.

a) At what time during the morning is the worker performing most efficiently? (NOTE: This is NOT maximizing Q(t), but the worker’s production rate R(t) = Q’(t)

b) At what time is the worker performing least efficiently?