For this case, the first thing we must do is define variables.
We have then:
x: unknown number 1
y: unknown number 2
The difference of both numbers is 124:
[tex] x - y = 124
[/tex]
The product of both numbers is:
[tex] P = x * y
[/tex]
Writing the product based on a variable we have:
[tex] P (x) = x * (x-124)
[/tex]
Rewriting:
[tex] P (x) = (x ^ 2-124x)
[/tex]
To find the minimum of the function, we must derive:
[tex] P '(x) = 2x-124
[/tex]
We equal zero and clear x:
[tex] 2x-124 = 0
2x = 124
x = 124/2
x = 62
[/tex]
Then, the value of the other number is given by:
[tex] y = x-124
y = 62-124
y = -62
[/tex]
Answer:
The searched numbers that meet both conditions are:
smaller number = 62
larger number = -62
