Question

Find the absolute maximum and minimum values of the function: [ f(x, y)=sqrt{(x-1)²(y+1)²] on the set D=left{(x, y) \mid x+y leq 4right). Hint: Use Lagrange multiplie

Answer 1

The absolute maximum and minimum values of the function f(x, y) on the set D can be found using Lagrange multipliers by solving the equation ∇f = λ ∇g where g(x, y) is the constraint. It's also important to check the boundary points for absolute extremum values.

Step-by-step explanation:

To find the absolute maximum and minimum values of the function f(x, y) = √((x-1)²(y+1)²) on the set D=(x, y) , we can use the method of Lagrange multipliers which involves setting the gradient of the function equal to the gradient of the constraint multiplied by a scalar λ, known as the Lagrange multiplier.

First, we identify the constraint g(x, y) = x+y - 4 = 0. Then, we calculate the gradients ∇f and ∇g and set ∇f = λ ∇g. Solving this system of equations will give critical points, which, along with any boundary points analyzed separately, can yield the absolute maximum and minimum values on the set D.

Remember to also check the behavior of the function on the boundaries of D by substituting y = 4 - x into f(x, y) and analyzing it as a single-variable function.