9 Answers

x+y=23
z=xy=y(23y)=23yy^2
dz/dy=232y=0 to be max
y=11.5
x=11.5

If you chearch 2 numbers whose you know them sum and them product, you can say that they are the solution of this equation :
x²  Sx + P = 0, where S is sum of them and P is product of them
You know sum : S = 23
x²  23x + P = 0
P =  x²  23x
To find a maximum (or minimum) of a function, you have to calculate the derivative of it.
P' =  2x  23
Then to get the result, solve the equation : P' = 0
P' = 0
 2x  23 = 0
2x =  23
x =  23/2
After this calculation, replace x by its value into the equation :
P =  x²  23x
P =  (23/2)²  23( 23/2)
P =  529/4 + 529/2
P =  529/4 + 1058/4
P = 529/4
Hen, you can write with the 2 numbers :
S = x + y = 23
P = xy = 529/4
You can deduce that : y = 23  x
You can substitute b by its value :
xy = 529/4
x(23  x) = 529/4
23x  x² = 529/4
x²  23x + 529/4 = 0
Polynomial like : ax² + bx + c, where :
a = 1
b =  23
c = 529/4
Δ = b²  4ac (discriminant)
Δ = ( 23)²  4(1 * 529/4) = 23²  529 = 529  529 = 0
x =  b / 2a = 23/2
But you know that :
x + y = 23
y = 23  x
y = 23  23/2
y = 23/2
The 2 numbers are : x = y = 23/2
Product : 529/4
Sum : 23
I'm French, sorry for language.

So we will let the two numbers be x and y.
Now we know how x and y relate. We have x + y = 23.
We want to maximize their product xy. But how do we do this without having a function of one variable?
Well we can solve our first equation for y to get y = 23  x.
Substituting this into our expression xy we get:
xy = x(23  x) = x^2 + 23x which we know is a "frowny face" parabola =). But this is good because we know that this type of parabola will have a maximum value! What is the maximum value? Or more specifically where does the value occur? It will occur at the vertex!
Using calculus, we take the derivative of x^2 + 23x to get 2x + 23 and set this equal to 0 to find where the graph has a horizontal tangent. It will occur at 2x + 23=0 => 2x = 23 => x=23/2
What is the value of the parabola at x = 23/2? it's just (23/2)^2 + 23(23/2) = 132.25+264.5 = 132.25
So we know that that the maximum value of the product will be 132.25 and the two numbers whose sum is 23 will be x = 23/2 and y = 23  (23/2) = 23/2
Good luck!

16 and 7

1+22=23 ......1*23=22
2+21=23 ........2*23=42
3+20=23...........2*23=60
so on
10+13=23 ........10*13=130
11+12=23..........11*23=132 [This is your max product] so the answer is 11 and 12
13+10=23..........13*23=130
After the max product has been reached the numbers will start going down and repeating it self.

I gues 11 and 12

11.5 and 11.5
If you mean whole numbers, it would be 11 and 12.

You could do this with calculus, but the quick way is to set them equal. 11.5*11.5 = 132.25

11,12