Final answer:
To find the values of x and y that maximize the product xy², we need to maximize x and minimize y. The values of x and y that maximize the product xy² are x = 2 and y = 10.
Step-by-step explanation:
To find the values of x and y that maximize the product xy², we need to maximize x and minimize y.
Since the sum of x and y is 12, we can set up an equation:
x + y = 12
Solving for y, we get:
y = 12 - x
Substituting this expression for y in the product xy²:
xy² = x(12 - x)² = 12x - x³
To find the maximum value of xy², we need to find the critical points of this function. Taking the derivative and setting it equal to zero:
dy/dx = 12 - 3x² = 0
3x² = 12
x² = 4
x = ±2
Since x and y are positive real numbers, we can choose x = 2 and solve for y:
y = 12 - x = 12 - 2 = 10
Therefore, the values of x and y that maximize the product xy² are x = 2 and y = 10.