Assume that a rectangular box has a diagonal of length $1 m$. What is the maximum possible volume of the box?
So I started by finding my function f which is: $$f(x,y,z) = xyz$$
Then I started to search for my constraint. We know that the diagonal of the box is $1$ and by projecting and using pythagorean theorem I got my constraint function $g$ to be: $$g(x,y,z) = x^2+y^2+z^2 = 1$$
Lastly I used Lagrange multipliers and got the correct answer: $$f\pm(\frac1{\sqrt{3}}, \frac1{\sqrt{3}}, \frac1{\sqrt{3}}) = \frac1{3\sqrt{3}}$$
I got correct answers (when I checkted the answer sheet, they just gave answers with no explanations) but I've previously solved a similar problem like this where they required me to also examine the fact that $x,y,z >0$ so am I done now? Or do I need to keeping going in which case how?