Page 228 - How To Solve Word Problems In Calculus

Step 1
Draw a diagram (if appropriate). Label all quantities,
known and unknown, which will be used in the problem.

Step 2
Write an equation representing the quantity to be max-
imized or minimized.

Step 3
Use any constraints or relationships between the vari-
ables to eliminate all but one independent variable. This
gives a function representing the quantity to be maximized
or minimized.

Step 4
Find all critical numbers. A critical number for f is a

number x for which either f (x) = 0or f (x) does not exist.

(In business problems it is extremely rare that f (x) will fail
to exist.)
Step 5
(optional) Use an appropriate test to confirm your abso-
lute maximum or minimum value (closed interval method,
first or second derivative tests—see Chapter 4). This step
may be omitted if you obtain only one critical number
and you are confident that the problem has a solution.

EXAMPLE 11

The wholesale price of a designer shirt is $25. A retail store
has determined that if they sell the shirt for $40, consumers
will purchase 55 shirts per month. The manager of the men’s
department knows that for each dollar decrease in price, 5
more shirts will be sold each month. What selling price will
yield the greatest monthly profit for the store?

Solution
Step 1
In this type of problem it is convenient to let x represent
the decrease in the price per shirt. The selling price is thus
40 − x dollars.

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