Let's say we have an inequality, then we change it into a function so for example we have:
$f(x,y) = x^2-xy-2$
Also we have the constraint
$g(x,y) = x+y - 1 = 0$
Using Lagrange multiplier, I get one point which is $$(x,y) = \left(\frac{1}{4}, \frac{3}{4}\right)$$
Obviously this function doesn't have a maximum, so this is the minumum point. But if we add one more constraint "$x,y \geq 0$," we should be able to obtain a maximum. How can we use this last constraint to obtain the maximum of the function?