4.7 Optimization Problems
These are applications in which we want to make a
given quantity as large or small as possible under a
set of constraints.
Example:
An open box is to be constructed from a piece of
cardboard 12 inches square by cutting squares of
equal size from each corner and folding up the
sides. Find the dimensions of the box of maximum
volume.
To Solve an Optimization Problem
1. Draw a diagram and label the given and
unknown quantities
2. Write an expression for the quantity to be
maximized or minimized
3. Re-express, if needed, in terms of 1 variable
4. Determine the domain
5. Find the absolute extremum by finding critical
numbers and using methods described in Ch. 4
6. Answer the question asked in a complete
sentence, including units
Closed Interval Test for Absolute Extrema
Let f be continuous on [a, b]. Then f has both an
absolute maximum and an absolute minimum on [a,
b]
Open Interval Test for Absolute Extrema
Suppose that c is a critical number for a continuous
function f on an interval.
If f has a local extremum at c on the interval and it is
the ONLY local extremum on the interval, then it is
an absolute extremum.
Example:
An open box is to be constructed from a piece of
cardboard 12 inches square by cutting squares of
equal size from each corner and folding up the
sides. Find the dimensions of the box of maximum
area.
Example 2.
A farmer has 2400 ft. of fencing and wants to fence
off a rectangular field that borders a straight river.
He needs no fence along the river. What are the
dimensions of the field that give the largest area?
Set up an equation for:
The sum of two positive numbers is 16. What is the
smallest possible value of the sum of their squares?
Ex. Determine the dimensions of an aluminum
can that holds 1000 cm3 (1 liter) of liquid and
that uses the least material. Assume that the
can is a cylinder, capped at both ends.