Use lagrange multipliers to find the max and min values of the func f(x,y)=xy subject to the constraint 1=(x^2)/8 + (y^2)/2

Question

I know how to set up..
i got y=Lx/4 and x=Ly

now im lost!!!
Please help with detail because im an idiot.

Answer 1

From the second equation, find
L=-x/y, substitute into the first to get
y-x^2/4y=0
which gives x=2y
Substitute in the third equation to get
4y^2/8+y^2/2-1=0
which gives y=1 which => x=2.
So (2,1) is a maximum/minimum.
Since the constraints have x^2 and y^2,
so
(-2,1), (-2,-1) and (2,-1) are all feasible solutions.

Locate the maxima and minima by substitution into F(x,y).

Answer 2

why is L=-x/y why is this negative??

Answer 3

L can be any number, positive, negative or even zero. It is present to ensure that the constraints are satisfied.

For example, we could have equally written the constraint equation as
L(1-x^2/8-y^2/2), then L would have been positive.