Question

#please help me

find the two positive numbers that satisfy the given requirements.
1. the sum is S and the product is a maximum

Answer 1

is this a calculus problem?

Answer 2

If their sum, S, is even, choose the two numbers to be S/2, since a square maximizes the area (the product of the two numbers) with a given perimeter.
If their sum is odd, then it can be written S=2n+1=n+(n+1), so let one number be n and the other n+1, since that's the closest to a square.

Answer 3

yes

Answer 4

let the 2 numbers be x and y
x+y= S
y= S-x

find the max of x*y = max of x(S-x)
You have a function f(x)= x(S-x)
to find its max, take the derivative, set = 0, and solve for x

Answer 5

Oh, I misread, thought it was only for integers.
Just let the smaller number be a and the bigger b, then you can write b as b=S-a. And use that to minimize the product P=a*b=a*(S-a).

Answer 6

**to find its max
to show you have a max (rather than a min),take the 2nd derivative and show it is negative.

Answer 7

how can i show it is negative? can you give the 1st and 2nd derivative?

Answer 8

\[ f(x)=x(S-x) = Sx-x^2\]
where S is constant.
Can you find the derivative?

Answer 9

S-2x

Answer 10

?

Answer 11

yes. and the 2nd derivative is -2, which means you will be finding a maximum when you solve
S-2x=0

Answer 12

what is the maximum?

Answer 13

you solve for x in
S-2x=0
you found the derivative of f(x). this is the slope. when the slope is 0, you are at a max