- #1

- 82

- 3

## 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?