Question

Find the minimum and maximum values of the function f(x,y)=x

2
y+x+y subject to the constraint xy=6. (Give exact answers. Use symbolic notation and fractions where needed. Enter DNE if the extreme value does not exist.)

Answer 1

The minimum value of the function f(x,y) =
`x^2y + x + y`subject to the constraint xy = 6 is -12, and the maximum value does not exist (DNE).

Explanation:

To find the extreme values of the function f(x,y) =
`x^2y + x + y`subject to the constraint xy = 6, we'll use the method of Lagrange multipliers. First, we express the constraint as g(x, y) = xy - 6 = 0. Then, we set up the Lagrangian function L(x, y, λ) = f(x,y) - λg(x,y) =
`x^2y + x + y`- λ(xy - 6 ).

Next, we find the partial derivatives of L with respect to x, y, and λ, setting them equal to zero to solve for critical points. Taking the partial derivatives:

∂L/∂x = 2xy + 1 - λy = 0,

∂L/∂y = x^2 + 1 - λx = 0,

∂L/∂λ = -xy + 6 = 0.

Solving this system of equations yields the values x = -2, y = -3, and λ = -1. However, this critical point does not satisfy the constraint xy = 6, so it's not valid.

Next, considering the constraint xy = 6, we notice that as xy = 6, the function f(x, y) becomes unbounded, without a maximum value. To determine the minimum, we note that the function
`x^2y` has no lower limit. As
`x^2y` can be made arbitrarily negative by selecting increasingly negative values of x and y satisfying xy = 6, the minimum value of f(x, y) is -12, achieved when x = -2 and y = -3, satisfying the constraint. Hence, the maximum value does not exist (DNE) due to the unbounded nature of the function.