Critical Points and the Second-Derivative Test

Extrema force the gradient to vanish

The Hessian discriminant classifies each candidate

Saddles are the new phenomenon

The test's silent case and its domain

How many critical points does $f(x, y) = x^3 - 3x + y^2$ have, and where are they?

For $f(x, y) = x^3 - 3x + y^2$ at the critical point $(-1, 0)$, the Hessian entries are $f_{xx} = -6$, $f_{xy} = 0$, $f_{yy} = 2$. What is the classification?

For the same function at the critical point $(1, 0)$, with $f_{xx} = 6$, $f_{xy} = 0$, $f_{yy} = 2$, what is the classification?

For $f(x, y) = x^2 - y^2$ at the origin, give the value of the discriminant $D$, as a plain number (negative numbers like -4 allowed).

For $f(x, y) = x^3 - 3x + y^2$, give the local minimum value, achieved at $(1, 0)$, as a plain number.