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Chapter 3.7–4
Find two positive numbers whose product is 154 and whose sum is a minimum.
Solution:
Let the two points be represented by the symbols x and y .
The phrase “two positive numbers whose product is 154” can be represented by the equation
xy = 154 . This is the equation that shows the relationship between x and y .
The phrase “whose sum is a minimum “ can be represented by the equation x + y = S , where S is a
real number. This is the equation where we minimize the sum.
From the equation xy = 154 , y =
154
154
. Substitute this value in the equation x +
=S.
x
x
Take the derivative, find the critical number(s), and test them.
x+
154
= x + 154 x −1 = S
x
S ' ( x ) = 1 − 154 x −2 = 0
x 2 = 154
x = ± 154
We delete the negative number because the problem stated positive numbers.
Let’s test the critical number. S ' (150 ) = 1 −
154
(
150 )
2
< 0 . This means the function is decreasing to the
left of the critical number.
S ' (160 ) = 1 −
154
(
160 )
2
> 0 This means the function is increasing to the right of the critical number.
This means that x = 154 is the location of a local minimum. The other number is
y=
154 154
=
= 154 .
x
154
The two numbers are both 154 . The product of the numbers is 154 and the sum is 2 154 .