How to Solve Lagrange Multiplier Problems for Function Extremes?

[tasveerk]

Homework Statement

Find the product of the maximal and the minimal values of the function
z = x - 2y + 2xy
in the region
(x -1)²+(y + 1/2)²≤2

Homework Equations

The Attempt at a Solution

I have taken the partial derivatives and set-up the problem, but I am having difficulty solving for x and y.
F(x,y) = x-2y+2xy-λ((x -1)²+(y + 1/2)² -2)
F_x = 1 - 2y - 2λ(x-1)
F_y = -2 + 2x - 2λ(y+1/2)
F_λ = -((x -1)²+(y + 1/2)² -2))

 

[Ray Vickson]

Homework Statement

Find the product of the maximal and the minimal values of the function
z = x - 2y + 2xy
in the region
(x -1)²+(y + 1/2)²≤2

Homework Equations

The Attempt at a Solution

I have taken the partial derivatives and set-up the problem, but I am having difficulty solving for x and y.
F(x,y) = x-2y+2xy-λ((x -1)²+(y + 1/2)² -2)
F_x = 1 - 2y - 2λ(x-1)
F_y = -2 + 2x - 2λ(y+1/2)
F_λ = -((x -1)²+(y + 1/2)² -2))

The first two equations are linear in x and y, so you can solve for x and y as functions of λ.

RGV