Why Must f(x,y,z) Attain a Maximum and Minimum on a Sphere?

[MuIotaTau]

Homework Statement

Explain why ##f(x,y,z) = x + y - z## must attain both a maximum and a minimum on the sphere ##x^2 + y^2 + z^2 = 81)##.

Homework Equations

None

The Attempt at a Solution

I know that any continuous function attains both a maximum and a minimum on a compact set. I defined the set ##K## as $$K = \{(x,y,z)|g(x,y,z) = 0\}$$ and demonstrated that it is bounded by a closed ball. I then realized, however, that I'm not really sure how to show that the set is closed. I understand the definition of a closed set, but I don't know how to show that it is true in this particular case. Any hints?

 

[pasmith]

Homework Statement

Explain why ##f(x,y,z) = x + y - z## must attain both a maximum and a minimum on the sphere ##x^2 + y^2 + z^2 = 81)##.

Homework Equations

None

The Attempt at a Solution

I know that any continuous function attains both a maximum and a minimum on a compact set. I defined the set ##K## as $$K = \{(x,y,z)|g(x,y,z) = 0\}$$

What is g?

and demonstrated that it is bounded by a closed ball. I then realized, however, that I'm not really sure how to show that the set is closed. I understand the definition of a closed set, but I don't know how to show that it is true in this particular case. Any hints?

It suffices to show that the sphere is compact. To do that, you can write down a continuous surjection from a compact subset of the plane to the sphere and use the result that the continuous image of a compact space is compact.

 

[Mark44]

Homework Statement

Explain why ##f(x,y,z) = x + y - z## must attain both a maximum and a minimum on the sphere ##x^2 + y^2 + z^2 = 81)##.

Homework Equations

None

The Attempt at a Solution

I know that any continuous function attains both a maximum and a minimum on a compact set. I defined the set ##K## as $$K = \{(x,y,z)|g(x,y,z) = 0\}$$ and demonstrated that it is bounded by a closed ball. I then realized, however, that I'm not really sure how to show that the set is closed. I understand the definition of a closed set, but I don't know how to show that it is true in this particular case. Any hints?

What's g(x, y, z)?

 

[MuIotaTau]

Oops, I'm sorry, I defined ##g## as ##g(x,y,z) = x^2 + y^z + z^2 - 81##. So the constraint equation.

We haven't demonstrated that compactness is continuous invariant in class, unfortunately, so I would be required to prove that. If demonstrating the set is closed is more complicated, I'll do it, but I would rather not prove a theorem if it's not strictly necessary.

 

[Dick]

Oops, I'm sorry, I defined ##g## as ##g(x,y,z) = x^2 + y^z + z^2 - 81##. So the constraint equation.

We haven't demonstrated that compactness is continuous invariant in class, unfortunately, so I would be required to prove that. If demonstrating the set is closed is more complicated, I'll do it, but I would rather not prove a theorem if it's not strictly necessary.

Proving that the set of points where g(x,y,z)=0 is closed should be easy. Just use the definition of continuity directly or that the inverse image of a closed set is closed for a continuous function. Now use Bolzano-Weierstrass.