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c
c
Kendra Kilmer February 28, 2012
Section 4.6  Optimization Problems
Strategy for Solving Optimization Problems
1. Introduce variables, look for relationships among these variables, and construct a mathematical model of the form
Maximize (or minimize)
f
(
x
)
on the interval
I
2. Find the critical values of
f
(
x
)
.
3. Use the procedures developed in previous sections (and below) to find the absolute maximum (or minimum) value
of
f
(
x
)
on the interval
I
and the value(s) of
x
where this occurs.
4. Use the solution to answer all questions asked in the problem.
Second Derivative Test for Absolute Extrema:
For a continuous function,
f
, on any interval, if
x
=
c
is the only
critical value in the interval where
f
′
(
c
) =
0 and
f
′′
(
c
)
exists, then
f
(
c
)
is
•
an
on
I
if
f
′′
(
c
)
>
0
•
an
on
I
if
f
′′
(
c
)
<
0
Note: The test fails if
f
′′
(
c
) =
0.
Example 1:
Find two positive numbers whose sum is 60 and whose product is a maximum.
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 Fall '08
 Allen
 Math

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