- #1
michonamona
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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
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) = [tex]\lambda[/tex]x/2
2. 2(1+x+y) = [tex]\lambda[/tex]y/8
3. x2/4 + y2/16 = 1
Then the textbook says that solving for [tex]\lambda[/tex] 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