# LaGrange Multiplier Problem

1. Mar 9, 2013

### Baumer8993

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

Consider the intersection of the elliptic paraboloid Z = X2+4Y2 , and the cylinder X2+Y2= 1. Use Lagrange multipliers to find the highest, and lowest points on the curve of intersection.

2. Relevant equations
The gradient equations of both functions.

3. The attempt at a solution

I have ∇f= <2X, 8Y> and ∇g= <2X, 2Y>. the constraint equation is X2+Y2 = 1.

I set the equations equal to get:

2X = (2X)λ 8Y = (2Y)λ

When I try to solve it always removes the variable. Where do I go from here to solve?

2. Mar 9, 2013

### Ray Vickson

Look at the first equation $2x = 2x \lambda$. Can you cancel the $2x$ on both sides? Why, or why not?

3. Mar 11, 2013

### Baumer8993

I think you can, but it would just leave me with λ = 1.

4. Mar 11, 2013

### Ray Vickson

OK, so? Work it through to the end.

BTW: there is another possibility having λ ≠ 1; can you see why?

5. Mar 11, 2013

### Baumer8993

I see that λ could also equal 4, but where do I go from plugging in the lambda values? I just end up with 2x=2x, and 8y=2Y, or do I need to use both λ lambda values at the same time?

6. Mar 11, 2013

### Ray Vickson

You have three equations, one involving x and λ, one involving y and λ, and the constraint.