# Constrained Optimisation

1. Oct 5, 2012

### notnottrue

1. The problem statement, all variables and given/known data
The temperature of a point on a unit sphere, centered at the origin, is given by
T(x,y,y)=xy+yz

2. Relevant equations
I know that the equation of a unit sphere is x^2+y^2+x^2=1, which will be the constraint.

3. The attempt at a solution
The partial derivatives of T are y, x+z and y respectively.
Unit circle partial derivatives are 2x, 2y and 2z.

From a theorem in the lecture notes∇T(x,y,z)=$\lambda$∇G(x,y,z)
G being the constraint. With the critical points when these equal 0.

So I get y-$\lambda$2x=0
x+z-$\lambda$2y=0
y-$\lambda$2z=0
with $\lambda$2x=$\lambda$2z

Firstly, am I on the right track? If so, what is the next move?
Thanks

2. Oct 5, 2012

### gabbagabbahey

Well, you haven't actually stated the question, so I would start with that...

3. Oct 5, 2012

### Ray Vickson

The last equality implies either (a) λ = 0; or (b) x = z. Try to see what else happens in both cases (a) and (b).

RGV