Page 89 - How To Solve Word Problems In Calculus

The first step in solving a maximum-minimum word
problem is to find a function that represents the quantity to be
maximized or minimized. Techniques for accomplishing this
task were discussed in Chapter 1 and are summarized below.

Step1
Draw a diagram (if appropriate). Label all quantities,
known and unknown, which are stated in the problem.
Step2
Write an equation representing the quantity to be max-
imized or minimized. This quantity will typically be rep-
resented in terms of two or more variables.

Step3
Use any constraints or relationships between the vari-
ables to eliminate all but one independent variable. This
converts the equation obtained in step 2 into a function.
Determine the domain of this function appropriate to the
problem; i.e., determine the set of all values of the inde-
pendent variable for which the problem makes sense.

Once the function has been found, we proceed to find its max-
imum or minimum value.

Step4
Find all critical numbers.
Step5
If the function is continuous on a closed interval, use
the closed interval method to determine its absolute max-
imum or minimum values.
OR
If there is only one critical value within the interval under
consideration, use the first or second derivative test to de-
termine whether it is a relative maximum or relative min-
imum. The value of the function at this location will be
the absolute maximum value or absolute minimum value,
respectively.

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