.

Taking the partial derivatives of

and setting them to

gives

and

.

Then my professor states that the critical points of

are

,

and

. Fine, I understand why, but I don't understand why the points

,

,

,

,

and

are not critical points.

How do you know that when you take

, then you have to chose

and not

and

?

Because I know of an example (

) which gives

and

. It still remains to solve for

and

(I see that

and

works for example. I've plotted the function and I saw 4 critical points (minimums), instead of 2 as my professor "would" find here.

So what am I overlooking?