Lagrange Multipliers

Okay.
That means both f(1,1) and f(-1,-1) are both minimums because fxx = 10 (which is greater than 0), and there are no maximums.

If you think about it this makes sense just by thinking about the equation. Both X and Y terms are squared so you will have identical positive and negative values. Ergo two minimum values.

Since there are no deducting terms (no subtraction signs), the values will only continue increasing. Ergo no maximum.

Can anyone explain what a lagrange multiplier is and how/why it works? I don't quite understand the textbook/wikipedia/google explanations.

Can't help you here unfortunately. Physics major. So long as the math works, I just go with it. They're used all over the place in matrix algebra...