Question

Find two numbers whose sum is 24 and whose product is the maximum possible value

Answer 1

To get the maximum value, the two numbers must be as close as possible.

24 ÷ 2 = 12

Both the numbers are 12.

Sum = 12 + 12 = 24
Max product = 12 x 12 = 144

Answer: The two numbers are 12 and 12.

Answer 2

let the numbers be x and y
thus
x+y=24
y=24-x
product is:
(x*y)=xy
P=x(24-x)
P=24x-x^2

For maximum value:
dp/dx=0
thus
dP/dx=24-2x=0
thus
24=2x
hence:
x=12
P is maximum at x=12, thus y=24-12=12