# Maximum and Minimum : Langrange multiplier problem

1. Nov 1, 2010

### michonamona

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

f(x,y) = (1+x+y)2

S = {(x,y) : x2/4 + y2/16 <= 1}

3. The attempt at a solution

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

fx(x,y) = 2(1+x+y)=0
fy(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) = $$\lambda$$x/2
2. 2(1+x+y) = $$\lambda$$y/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

2. Nov 1, 2010

### 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.