Find the minimum of \(\displaystyle x^2-2x+y\) if \(\displaystyle x,y \ge 0\) and \(\displaystyle x+y \le 2\). I need a graphic interpretation of the problem (in the plane xOy). Thank you!

There is no minimum. For example, if you fix \(\displaystyle x=0\), then \(\displaystyle k=x^2-2x+y=y\) and then, since you only need \(\displaystyle y\leq2\), you can take \(\displaystyle y\), and hence \(\displaystyle k\), arbitrarily close to \(\displaystyle -\infty\).

There is no maximum either. For example, if you fix \(\displaystyle y=0\), then \(\displaystyle k=x^2-2x+y=x^2-2x\) and then, since you only need \(\displaystyle x\leq2\), taking \(\displaystyle x\) arbitrarily close to \(\displaystyle -\infty\) will make \(\displaystyle k\) arbitrarily close to \(\displaystyle +\infty\).