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The product of two numbers is 72. The sum of one number and 2 times the second is to be minimized. Determine the two numbers.We need variables for the two numbers, a and b. The first condition can be stated as
.
The condition on the sum, S, can be stated as
.
To express the minimum condition, we need to express S in terms of one variable and differentiate. We can solve the first equation to get
.
We can substitute this into the expression for S :
.
Now we can differentiate to get the derivative for any value of b:
.
To find the minimum at bmin : 
,
or 
,
or 
Now we can use this result in the definition of a to find the corresponding value of a:
So the two numbers are 6 and 12.
As a check, we can calculate S for several pairs of nearby values:
|
a |
b |
S |
|
12 |
6 |
24 |
|
14.4 |
5 |
24.4 |
|
10.3 |
7 |
24.3 |
|
11 |
6.55 |
24.1 |
|
13 |
5.54 |
24.1 |