# Find two numbers whose sum is 23 and whose product is a maximum?

• x+y=23

z=xy=y(23-y)=23y-y^2

dz/dy=23-2y=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