I know these type of question have been asked here before but I was actually confused with a hint my professor gave me regarding the following question:
Let $D = \{(x,y,z) \in \mathbb R^3 : x + y + z = 1\}$, a plane in $\mathbb R^3$, and let $f:\mathbb R^3 \to \mathbb R$ be a function defined as $f(x,y,z) = x^2 + 2y^2+3z^2$. Does $f$ have a minimum and a maximum in $D$?
Usually when the constraint is a compact set (when it's closed and bounded) we can conclude $f|_D$ does have a min and a max when $f$ is continuous. Here obviously the plane isn't bounded so we can't say that.
I did conclude $f$ has a min by taking a compact set $K$ of the plane and showing that the minimum $f$ has in $D \cap K$ is a global minimum. But how do you show that $f$ has or doesn't have a maximum? My professor hinted that $f(x) \ge \Vert x \Vert ^2$ for all $x \in R^3$. I understand $\Vert x \Vert^2 $ is unbounded, but I'm not sure how it is unbouded in $D$ and how this implies $f$ is also unbounded on $D$.