The same proposition is valid for a function of two variables, when the constraint is given by a curve in the plane.
If x=0, then and we found two points and .
If , then z=0 and and this satisfies the first equation too. We get two points P_{3}(1,0,0) and P_{4}(-1,0,0).
Which of these points correpsond to minima and which of them correspond to maxima? We compute the values:
We compute the gradients:
Using Prop. 6.28, we have: