Lagrange Multipliers

[terminal.velo]

Homework Statement

Find max/min of f subject to constraint: x^2+y^2+z^1 = 1

Homework Equations

f(x,y,z) = 1/4*x^2 + 1/9*y^2 + z^2
g(x,y,z) = x^2 + y^2 + z^2 - 1

The Attempt at a Solution

L = 1/4*x^2 + 1/9*y^2 + z^2 - λ(x^2 + y^2 + z^2 - 1)
Lx = 2/4*x - λ*x*2
Ly = 2/9*y - λ*2*y
Lz = 2*z - λ*2*z

 

[ehild]

Homework Statement

Find max/min of f subject to constraint: x^2+y^2+z^1 = 1

Homework Equations

f(x,y,z) = 1/4*x^2 + 1/9*y^2 + z^2
g(x,y,z) = x^2 + y^2 + z^2 - 1

The Attempt at a Solution

L = 1/4*x^2 + 1/9*y^2 + z^2 - λ(x^2 + y^2 + z^2 - 1)
Lx = 2/4*x - λ*x*2
Ly = 2/9*y - λ*2*y
Lz = 2*z - λ*2*z

Correct so far. What should be the partial derivatives at a minimum/maximum?

ehild

 

[terminal.velo]

I don't know, the lecturer told us to solve the rest of the problem at home.

 

[ehild]

Well, check your notes, what is the condition that a minimum or maximum exist?

ehild

 

[terminal.velo]

All 3 derivates of L (Lx, Ly, Lz) have to equal zero, to find the λ, then the x/y/z for critical points of f(x,y,z)

Lx = 2/4*x - λ*x*2 = 0
Ly = 2/9*y - λ*2*y = 0
Lz = 2*z - λ*2*z = 0

 

[ehild]

OK, factorize the left hand sides.

ehild

 

[terminal.velo]

2x(1/4 - λ) = 0
2y(1/9 - λ) = 0
2z(1 - λ) = 0

EDIT: I'm in a bit of a funk, apologies! (lectures from 8am 'til 3pm)

 

[ehild]

2x(1/4 - λ) = 0
2y(1/9 - λ) = 0
2z(1 - λ) = 0

EDIT: I'm in a bit of a funk, apologies! (lectures from 8am 'til 3pm)

So you need some rest, without Maths...
Anyway: you can omit the factors 2. All your equations are products, equal to zero. That means, one of the two factors must be zero in each equation. More than one factor containing lambda can not be zero, it would mean contradiction. Can all x, y,z equal to zero? Remember, you have the condition that x²+y²+z²=1.

ehild