Find two positive number that satisfies the given requirements

Question

Question: The sum of the first number squared and the second number is 54 and the product is a maximum.

Answer 1

x^2 + y = 54

y = 54 - x^2

The product is x(54 - x^2) = 54x - x^3 = f(x)

0 = f'(x) = 54 - 3x^2

0 = 18 - x^2

x^2 = 18

x = 3*Sqrt(2), as x>0

x=3*Sqrt(2) and y = 36

the max product is 108*Sqrt(2)