临界点与二阶导数判别法

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.