Can You Find Two Numbers Whose Sum Is 72 And Whose Product Is A Maximum?

Anonymous answered

We impose that the first number =X
the second =y
the maximum product=m(x)
y+x=72
y=72-x
M(x)=x * y
M(x)=x *(72-x)
M(x)=72x- X^2 (by deriving the both sides of the equation)
M'(x)=72-2X
we Put M' (x)=0
72 - 2X=0
2X =72
X= 36 Y =72 -36
Y =36

Oddman answered

36 and 36 are the numbers you seek.

An equation for the product as a function of the smaller number (n) is
p(n) = n(72-n)
p(n) = -n² + 72n
This quadratic expression has a maximum at n = -72/(2*(-1)) = 36
Check 36*36 = 1296
(36 - .01)(36 + .01) = 36*36 - .01*.01 = 1295.9999
You can see that the square of any deviation from 36 is subtracted from the square of 36. Thus, the product 36*36 is as large as you can get.

Can You Find Two Numbers Whose Sum Is 72 And Whose Product Is A Maximum?

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