Solving practical problems that ask us to maximize or minimize a
quantity are typically called optimization
problems in calculus. These problems occur perhaps
more than any others in the real world (of course, our versions
used to teach these methods are simpler and contrived.)

One of the main reasons we learned to find maximum and minimum
values in the previous sections was so that we could use this
skill to optimize functions. Keep
in mind: we find the maximum or minimum value of a
function by differentiating, finding critical numbers, and
determining if these yield extreme values. At
the heart of each of these optimization problems is
differentiation.

Strategy

The videos on this and following pages will illustrate different
examples where we are asked to minimize or maximize a quantity.
Make sure to pay attention to constraints that are naturally
implied in the problem and to follow the strategy outlined below:

Process:

Determine that this is an optimization problem
(wording includes words like "largest," "smallest,"
"greatest," "least," etc.).

Draw a picture (as always when working with
word problems)

Identify what is known and unknown, and
assign variables to the unknown quantities.

Determine what value needs to be optimized
(maximized or minimized).

Find a function that models the value to be
maximized or minimized.

If this function has two variables, use additional
information in the problem to eliminate one of the
variables. To do this, find a relationship
between the two variables, usually given by some
constraint in the problem, and solve for one variable in
terms of the other.

Find the absolute extreme value(s) of the
function of one variable, as you have learned to do
previously.

Determine the answer by re-reading the
question, and using your extreme value(s) found to find
the answer.