Question

Find two positive real numbers whose product is a maximum.
The sum is A.

Answer 1

is it be A/2 and A/2

Answer 2

let the numbers be x and A-x
hence their product =x(A-x)=xA -x^2=(A^2)/4 -(A^2)/4 +2xA/2 -x^2
=(A^2)/4 -(x-A/2)^2
thus the maximum value of the product is (A^2)/4 when the -ve part is 0 i.e when x=A/2
thus one number is A/2 and thus the other one is A-A/2=A/2

Answer 3

\[xy= \max\]

Answer 4

as A is a constant

Answer 5

x + y = A
y = A -x

Answer 6

hence we can write
xy = max
as
x*(A-x) = max

Answer 7

xA - x2 = max

Answer 8

now differentiating it
A - 2x = 0
A = 2x

Answer 9

and x = A/2

Answer 10

and y = A - x
y = A - A/2
y = A/2