How Do You Solve a Constrained Optimization Problem on a Unit Sphere?

[notnottrue]

Homework Statement

The temperature of a point on a unit sphere, centered at the origin, is given by
T(x,y,y)=xy+yz

Homework Equations

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

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)=[itex]\lambda[/itex]∇G(x,y,z)
G being the constraint. With the critical points when these equal 0.

So I get y-[itex]\lambda[/itex]2x=0
x+z-[itex]\lambda[/itex]2y=0
y-[itex]\lambda[/itex]2z=0
with [itex]\lambda[/itex]2x=[itex]\lambda[/itex]2z

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

 

[gabbagabbahey]

Homework Statement

The temperature of a point on a unit sphere, centered at the origin, is given by
T(x,y,y)=xy+yz

Homework Equations

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

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)=[itex]\lambda[/itex]∇G(x,y,z)
G being the constraint. With the critical points when these equal 0.

So I get y-[itex]\lambda[/itex]2x=0
x+z-[itex]\lambda[/itex]2y=0
y-[itex]\lambda[/itex]2z=0
with [itex]\lambda[/itex]2x=[itex]\lambda[/itex]2z

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

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

 

[Ray Vickson]

Homework Statement

The temperature of a point on a unit sphere, centered at the origin, is given by
T(x,y,y)=xy+yz

Homework Equations

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

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)=[itex]\lambda[/itex]∇G(x,y,z)
G being the constraint. With the critical points when these equal 0.

So I get y-[itex]\lambda[/itex]2x=0
x+z-[itex]\lambda[/itex]2y=0
y-[itex]\lambda[/itex]2z=0
with [itex]\lambda[/itex]2x=[itex]\lambda[/itex]2z

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

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