Finding the maximum and minimum of $f$ on a set $Q$ I stumbled across a proposed task that I'm unable to solve.
We have a function $f:\mathbb{R^2}\to\mathbb{R}$ defined as:
$$
f(x,y):=x^2+xy+y^2+x+y+1
$$
The task is to "find the maximum and minimum" of $f$ on the set $Q$ which is defined as:
$$
Q:=\{(x,y)\in\mathbb{R^2}|\;\|(x,y)\|_\infty\le1\}
$$
Not knowing how to proceed, I decided to ignore $Q$ and simply calculated possible critical points and extrema. $f$ turns out to have a strict local minimum at $\left(-\frac13,-\frac13\right)$ which would fit into the constraints of $Q$ as $\|\left(-\frac13,-\frac13\right)\|_\infty$ is $\left(\frac13\right)\le1$.
How am I to proceed from this point on? Am I mistaken in thinking that the same strict local minimum is kept and that there is no maximum?
Cheers!

There must be a maximum,as $Q$ is compact. Your method gives you all the local extrema in the interior of $Q$. So you need to check the values on the boundary of $Q$ next. 
That means, you have to compute the extreme values of $x \mapsto f(x,\pm 1)$ and $y \mapsto f(\pm 1, y)$ on $[-1,1]$ and compare them to the inner extremum you allready found.
