To find the extremum of f(x, y) = xy subject to the constraint 4x + y = 18, we can use the method of Lagrange multipliers.
First, let's define the Lagrangian function L(x, y, λ) as:
L(x, y, λ) = xy + λ(4x + y - 18)
Next, we need to find the critical points by taking the partial derivatives of L with respect to x, y, and λ, and setting them equal to zero:
∂L/∂x = y + 4λ = 0 -- (1)
∂L/∂y = x + λ = 0 -- (2)
∂L/∂λ = 4x + y - 18 = 0 -- (3)
From equations (1) and (2), we can solve for x and y in terms of λ:
y = -4λ -- (4)
x = -λ -- (5)
Substituting equations (4) and (5) into equation (3), we can solve for λ:
4(-λ) + (-4λ) - 18 = 0
-8λ - 18 = 0
λ = -18/8
λ = -9/4
Substituting λ = -9/4 back into equations (4) and (5), we can find the corresponding values of x and y:
y = -4(-9/4) = 9
x = -(-9/4) = 9/4
So, the critical point (x, y) is (9/4, 9) with λ = -9/4.
To determine whether this critical point is a maximum or a minimum, we need to evaluate the second-order partial derivatives of f(x, y) = xy.
The second-order partial derivatives are:
∂²f/∂x² = 0
∂²f/∂y² = 0
∂²f/∂x∂y = 1
Now, let's calculate the determinant of the Hessian matrix:
H = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)²
= (0)(0) - (1)²
= -1
Since the determinant of the Hessian matrix is negative, the critical point (9/4, 9) corresponds to a saddle point. Therefore, there is no maximum or minimum for the function f(x, y) = xy subject to the given constraint 4x + y = 18.