# Lagrange Multipliers

1. Jul 22, 2010

### Quincy

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

Find the points on the ellipse x2 + 2y2 = 1 where f(x,y) = xy has its extreme values.

2. Relevant equations

3. The attempt at a solution
f(x,y,z) = x2 + y2 + z2 -- constraint
g(x,y,z) = x2 + 2y2 -1 = 0

gradient of f = $$\lambda$$ * gradient of g

2xi + 2yj + 2zk = $$\lambda$$2xi + $$\lambda$$4yj

2x = $$\lambda$$2x
2y = $$\lambda$$4y
2z = 0

I don't know where to go from here, can someone help me out?

2. Jul 22, 2010

### Quincy

Sorry, I made a mistake in my original post, it should be:

f(x,y,z) = xy
g(x,y,z) = x2 + 2y2 - 1

yi + xj = $$\lambda$$(2xi + 4yj)
y = 2x$$\lambda$$
x = 4y$$\lambda$$

y = 2(4y$$\lambda$$)$$\lambda$$ = 8y$$\lambda$$2

$$\lambda$$ = sqrt(1/8)

where do i go from here?

3. Jul 22, 2010

### Office_Shredder

Staff Emeritus
you have three equations and three unknowns.... the two gradient equations and the fact that x2+2y2=1. Now that you have solved for one unknown, can you make it two equations with two unknowns?

4. Jul 22, 2010

### Quincy

Find the minimum distance from the surface x2 - y2 - z2 = 1 to the origin. (function being minimized = x2 + y2 + z2)

2xi + 2yj + 2zk = $$\lambda$$(2xi - 2yj -2zk)

2x = 2x$$\lambda$$
2y = -2y$$\lambda$$
2z = -2z$$\lambda$$
x2 = y2 +z2 + 1

- I can't figure out how to solve this system of equations, any tips?

5. Jul 23, 2010

### HallsofIvy

Staff Emeritus
These are pretty close to being trivial. From the first equation, if $x\ne 0$, $\lambda$ must be 1. From the second equation, if $y\ne 0$, $\lambda$ must be -1, and from the third, if $z\ne 0$, $\lambda$ must be -1.

Since $\lambda$ cannot be both 1 and -1, at least one of the coordinates must be 0!

Suppose x= 0. Then the constraint becomes $-y^2- z^2= 1$ which is impossible. If x is not 0, then $\lambda$ must be 1, not -1, so y and z must be 0. The constraint becomes $x^2= 1$.

Looking at it geometrically gives an easy check. This is a "hyperboloid of two sheets" with the x-axis as axis of symmetry.

Last edited: Jul 23, 2010