Flat spots: peaks, pits, and passes

Every optimum — cheapest, strongest, most profitable, best-fitting — lives at a special kind of place: where the landscape goes flat. Finding those places is the easy half of optimization. Telling them apart is the art.

Where the gradient dies

At a summit you can't climb any higher — no direction goes up. At the bottom of a valley, none goes down. In both cases the steepest-ascent arrow has nowhere to point:

$\nabla f = (0, 0)$

A point where the gradient vanishes is a critical point. Finding them is a system of two equations — set both partials to zero — algebra you already own (it's module 2 of the algebra course, in costume).

Flat is not the same as extreme

Here's the trap. Three very different terrains all read ∇f = 0:

• Maximum — a summit: every direction descends.
• Minimum — a pit: every direction climbs.
• Saddle — a mountain pass: uphill along one axis, downhill along the other. Flat at the center, extreme in no sense.

The saddle is the new character — single-variable calculus never met one, because with one direction there's no way to disagree. With two directions, the x-slice and the y-slice can vote opposite ways.

Worked: find them by hand

f(x, y) = x³ − 3x + y². Partials: ∂f/∂x = 3x² − 3 and ∂f/∂y = 2y. Setting both to zero: x = ±1, y = 0 — two critical points, (1, 0) and (−1, 0). Which is which? Near (1, 0) the surface behaves like a pit; near (−1, 0), like a saddle. You can check that by poking nearby values… but poking isn't a proof.

See all three characters as terrain in the surface explorer — the bowl, the dome (bowl with a negative), and the saddle family, each with its origin classified for you.

Why this matters

∇f = 0 gives you candidates; it can't rank them. The final lesson builds the classifier — and the tool it uses is the eigenvalue machinery you built in module 3. The two halves of this course are about to meet.