Find the two real numbers whose sum is 4 and whose product is a maximum.

Let the two real numbers be x and y.

Their sum is given by:

x + y = 4

Then, you are told that their product is a maximum.

Hence,

xy = k

Where I'll call k the maximum value.

To get a maximum, x and y should both be positive (or both negative, but since their sum is positive, they cannot be both negative)

What are the numbers which add up to 4?
0 and 4
1 and 3
2 and 2

Product:
0
3
4

I think you got your answer now :)

x+y = 4 from the first part.
the product is f = xy
substituting,
f = x(4–x) = 4x – x²
To maximize this, differentiate and set equal to 0 to find the max/min
f' = 4 – 2x = 0
2x = 4
x = 2
y = 2

b