1. The Extremum Question — Upgraded to Several Variables
Callback — Single-Variable Definitions
Recall from the single-variable block:
- A local maximum of $f$ at $x_0$ means $f(x_0) \ge f(x)$ for every $x$ in some open interval around $x_0$.
- A local minimum of $f$ at $x_0$ means $f(x_0) \le f(x)$ for every $x$ in some open interval around $x_0$.
We now lift these definitions verbatim into two (or more) variables: replace "open interval" with "open disk" (or open ball).
Definition — Local Extrema in Several Variables
Let $f(x,y)$ be defined on a region $D \subseteq \mathbb{R}^2$. We say $f$ has a local maximum at $(a,b)$ if there exists an open disk centred at $(a,b)$ such that $f(a,b) \ge f(x,y)$ for every $(x,y)$ in that disk. A local minimum is defined analogously with $\le$.
The New Citizen: Saddle Points
In one variable, a critical point is either a local max, a local min, or an inflection point. In two variables a genuinely new phenomenon appears — the saddle point. A saddle point is a critical point where the function behaves like a minimum in one direction and like a maximum in another. The classic witness is:
$$f(x,y) = x^2 - y^2$$At the origin both partial derivatives vanish ($f_x = 2x = 0$, $f_y = -2y = 0$), yet the origin is neither a local max nor a local min. Along the $x$-axis the surface curves upward; along the $y$-axis it curves downward. This is the standing witness that "critical does not mean extremal."
Necessary Condition — Fermat's Theorem Upgraded
Theorem (Fermat, multivariable). If $f$ has a local extremum at an interior point $(a,b)$ and $f$ is differentiable there, then