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Method #1: Study the sign of
xvalues sample number . and qualify is the location of a max or a min or neither. There are two ways to do that:
on the interval
.
sign of conclusion about
is decreasing
has horizontal tangent line
is increasing We see that on the interval
the graph of is decreasing until
, then increasing after
. So the
critical value
is the location of the absolute min for the interval
. But remember the absolute min is
the yvalue, not the xvalue. We substitute
into
to get a yvalue. The result:
. Conclude that the absolute min is . (It occurs at .) Method #2: Study the sign of
the interval
.
So far, we have found that
. Therefore,
. We see that
for all
, the value of
will be positive. So the graph of must be concave up for all
. So the
critical value
is the location of the absolute min for the interval
. As we did in the previous method,
we find the value of the absolute min to be
. [5] (suggested exercise 56#11) A company manufactures and sells hats. The demand is the number of hats made each day. The price is the selling price for each hat (in dollars). The daily pricedemand equation is
. Remember that
. The Cost function is
(in dollars). Remember that
.
(A) Graph the pricedemand equation and find its domain. Explain how you know the domain.
Solution: The graph of the equation
will be a line with
slope
. The vertical axis intercept will be at the point
; the horizontal axis intercept will be at the point
.
If we considered the equation
as an abstract mathematical
equation, its domain would be the set of all real numbers . But the
equation is not just an abstract mathematical equation: it is modeling
the making and selling of hats. Because one cannot make a negative
number of hats, we know that is restricted to
. And because
price cannot be negative...
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This document was uploaded on 02/17/2014 for the course MATH 1350 at Ohio University Athens.
 Fall '12
 barsamnian
 Math, Algebra

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