Maximum and Minimum : Langrange multiplier problem

[michonamona]

Homework Statement

Find the maximum and minimum of the function f over the closed and bounded set S. Use langrange multiplier method to find the values of the boundary points.

Homework Equations

f(x,y) = (1+x+y)²

S = {(x,y) : x²/4 + y²/16 <= 1}

The Attempt at a Solution

First, I set their first order partial derivatives to 0 to get the following

f_x(x,y) = 2(1+x+y)=0
f_y(x,y) = 2(1+x+y)=0.

It's obvious that I'm not going to be able to find a unique value for my critical points with these two equations, thus, I conclude that there are infinitely many critical points in the interior (I also don't understand the intuition behind this conclusion).

Next, we check the points that satisfy x²/4 + y²/16 = 1, to see if these are potential extremums. I will use the langrange multiplier method.

1. 2(1+x+y) = \lambdax/2
2. 2(1+x+y) = \lambday/8
3. x²/4 + y²/16 = 1

Then the textbook says that solving for \lambda will yield y=-x-1 or y=4x. I know how to get y=4x, but where did y=-x-1 come from? How were they able to derive it from these three equations?

Thank you very much for your help,

M

 

[MathematicalPhysicist]

Your first case is for lambda different than zero, so you can identify the first two equations and then divide by lambda.
If lambda equals zero, then you get 1+x+y=0, which is the second solution.