What do I do if they ask me to find all critical points of the form (x, 0) for this function : $f(x, y) = xe^y + x^3 − y^2 − 4x − xy.$

Question

I am unsure if I need to set y = 0 and then try to find the critical points and classify them, or do I take the derivates and then I set y = 0 and then try to find the critical points and classify them? Does it change something? I did it both ways, and it gave me the same answer. the only difference was for the first try, when I computed the determinant of the Hessian, it gave me 0, so I only knew if it was a max or a min by doing the second derivative test only with respect to x. Was it a coincidence that the answer was the same or should i always do it the second manner?

Answer 1

If you set $y = 0$ before taking the derivative, then you're really looking at the critical points of the univariate function $f(x) = x^3 - 3x$. From that, you can't actually guarantee that the derivative in the $y$ direction is actually going to be zero, meaning that you won't know whether you have a critical point in the first place (in this situation $\frac{\partial f}{\partial y}$ is zero along the line $y = 0$ so that isn't an issue, but that's far from guaranteed).

Answer 2

For the first case where you set $y = 0$, the function becomes a function of a single variable; therefore, the Hessian matrix would be $1 \times 1$, i.e. the second derivative test only with respect to $x$. Both methods are valid, it's a matter of preference which one you choose. Personally, the first method is preferred because it requires much less computation.