Question

The function f(x,y) = 3xy has an absolute maximum value and absolute minimum value subject to the constraint x² + y² - xy = 9. Use Lagrange multipliers to find these values. Find the gradient of f(x,y) = 3xy. Vf(x,y) =

Answer 1

To find the absolute maximum and minimum values of the function f(x,y) = 3xy subject to the constraint x² + y² - xy = 9, we can use Lagrange multipliers. The gradient of f(x,y) = 3xy is given by ∇f(x,y) = (3y, 3x).

Step-by-step explanation:

To find the absolute maximum and minimum values of the function f(x,y) = 3xy subject to the constraint x² + y² - xy = 9, we can use Lagrange multipliers. First, we need to define the function L(x, y, λ) which is the sum of the function f(x,y) and the constraint equation multiplied by the Lagrange multiplier λ. So, L(x, y, λ) = 3xy + λ(x² + y² - xy - 9).

To find the critical points, we need to find the partial derivatives of L(x, y, λ) with respect to x, y, and λ, and set them equal to zero. Solving the resulting equations will give us the points where the absolute maximum and minimum can occur. After solving the system of equations, we can find the values of x and y that correspond to the extreme values of the function f(x,y) = 3xy.

The gradient of f(x,y) = 3xy is given by ∇f(x,y) = (3y, 3x), where ∇ denotes the gradient operator. This means that the partial derivative of f with respect to x is 3y, and the partial derivative of f with respect to y is 3x.