Example
1.
Find
two nonnegative numbers whose sum is 9 and so that the product of one number
and the square of the other number is a maximum.
Solution: Let variables x and y represent two nonnegative
numbers. The sum of the two numbers is given to be 9 = x + y
so that
y = 9 - x
We wish to MAXIMIZE the PRODUCT
P = x y2 .
However, before we differentiate the
right-hand side, we will write it as a function of x only. Substitute for y getting
P = x y2
= x ( 9-x)2
Now differentiate this equation using
the product rule and chain rule, getting
P' = x (2) ( 9-x)(-1)
+ (1) ( 9-x)2
= ( 9-x) [ -2x +
( 9-x) ]
= ( 9-x) [ 9-3x ]
= ( 9-x) (3)[ 3-x ]
= 0
for
x=9 or x=3 .
Note that since both x and y are nonnegative numbers and
their sum is 9, it follows that 0 ≤ x ≤ 9.
If
|
and
|
then
|
x=0
|
y=9
|
P=0
|
x=3
|
y=6
|
P=108
|
x=9
|
y=0
|
P=0
|
If x=3 and y=6, then P=108 is the
largest possible product. (substitute to P=xy2)
Example
2.
A rectangular page is to contain 24 square inches of
print. The margins at the top and bottom of the page are to be 1 
inches, and the margins
on the left and right are to be 1 inch. What should the dimensions of the page
be so that least amount of paper is used?
inches, and the margins
on the left and right are to be 1 inch. What should the dimensions of the page
be so that least amount of paper is used?
Solution: Let A be the area to be
minimized
A = (x+3)(y+2) primary equation
The printed area inside the margins is
given by
24 = xy secondary
equation
Solving this equation for y produces y= 
. Substituting into the primary equation produces
. Substituting into the primary equation produces
A = (x+3)(+2)
+2)
=30+2x+
function of
one variable
function of
one variable
Because x must be positive, you are interested only in values of A for x>0. To find the critical
numbers, differentiate with respect to x.
= 2-
= 0 or x2=36
So the critical numbers
are x=±6. You do not have to consider x=-6 because it is outside the domain.
The First Derivative Test confirms that A is a minimum when x=6. So, y=
=4 and the dimensions of
the page should be x+3=9 inches by y+2=6 inches.
=4 and the dimensions of
the page should be x+3=9 inches by y+2=6 inches.
Calculus Concepts and Contexts Second Edition by James Stewart
You're a genius Sarge! ======D
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