Critical points — finding, classifying: Foundations

This page assumes nothing. If the parent note used a symbol and expected you to already know it, we build it here from a picture first. Read top to bottom — each idea is the ground the next one stands on.

0. The picture everything lives in

Before any symbol, fix the mental image. A function of two variables $f (x, y)$ is a height machine: you feed it two numbers (a spot on a flat map, like east-position $x$ and north-position $y$ ) and it hands you back one number — the height of the land at that spot. We call that height $z$ .

Figure — Critical points — finding, classifying

Why the topic needs it: critical points are special spots on the floor $(x, y)$ where the surface above does something interesting (peak, pit, pass). No surface, no story.

1. The coordinates $x, y$ and the point $(a, b)$

The two inputs are named $x$ and $y$ . A specific spot we care about often gets its own letters, usually $(a, b)$ — think of it as "we planted a flag at east-position $a$ , north-position $b$ ."

Why: the parent writes things like " $(a, b)$ is a critical point" — it is asking a yes/no question about one dot.

2. Slope in one direction: the partial derivative $f_{x}$

Stand on the surface. Now walk only east (increase $x$ , keep $y$ frozen). You trace a single curve — a slice of the surface. The steepness of that slice is a slope. Freezing $y$ and measuring the east-west slope gives us the partial derivative with respect to $x$ , written $f_{x}$ .

Why the topic needs it: a flat spot must be flat east–west AND north–south. That is exactly the two conditions $f_{x} = 0$ and $f_{y} = 0$ . See Gradient and Directional Derivatives for the full directional story.

3. Bundling both slopes: the gradient $\nabla f$

Two slopes $f_{x}$ and $f_{y}$ describe the ground fully. We staple them into one arrow called the gradient, written $\nabla f$ (the symbol $\nabla$ is read "nabla" or "del" — just a name for "collect the partial slopes").

Why: the parent's whole definition of a critical point is one line — $\nabla f (a, b) = 0$ . Now you can read it: "the steepest-uphill arrow has shrunk to nothing, because there's no uphill — you're on a flat spot."

4. Second slopes: curvature $f_{xx}, f_{y y}, f_{x y}$

At a flat spot the arrow is gone, so slope tells us nothing about shape. We need how the slope itself is changing — that is curvature. Take a derivative of a derivative.

$f_{xx}$ = "how the east-slope changes as you keep walking east" = up-curve or down-curve along $x$ . Positive = bowl-shaped (cup 🙂), negative = dome-shaped (cap 🙁).
$f_{y y}$ = same idea along the north direction.
$f_{x y}$ = the mixed one: "how does the east-slope change as you step north?" It measures how the two directions twist together.

Why the topic needs it: these three numbers are the raw material of the Hessian Matrix and the whole Second Derivative Test.

5. Packaging curvature: the Hessian matrix $H$ and $det H$

Three curvature numbers deserve a tidy box. A matrix is just a rectangular grid of numbers. The Hessian $H$ is the 2×2 grid of second partials.

Why: the parent's boxed formula $D = det H$ is now not magic — it's "multiply the straight curvatures, subtract the twist squared."

6. Reading the sign of a quadratic: the quadratic form $Q$

Near a flat spot, the change in height for a tiny step $(h, k)$ is (approximately) a quadratic form — a sum of squared-ish terms: $Q (h, k) = \frac{1}{2} (f_{xx} h^{2} + 2 f_{x y} hk + f_{y y} k^{2}) .$

Here $h, k$ are just small step sizes east and north. "Quadratic" means every term is degree-2 (each variable appears twice: $h^{2}$ , $hk$ , $k^{2}$ ).

Why: this quadratic is the bridge, built from the Taylor Series in Several Variables, between the raw curvature numbers and the plain-English verdict.

7. The tool that makes $Q$ appear: Taylor expansion & " $\approx$ "

How do we know the height change looks quadratic near a flat spot? Because a Taylor expansion approximates any smooth surface near a point by a polynomial: constant + linear + quadratic + tinier corrections.

At a critical point the constant part is just the height itself and the linear part is zero (slopes gone), leaving $Q$ as the leading actor. The 1-D warm-up for this whole idea is the Single-variable Second Derivative Test; constraints get handled separately by Lagrange Multipliers.

How these feed the topic

Equipment checklist

Read the left side, answer, then reveal.

What does

f (x, y)

output, and what shape do all its outputs draw?

One height number

z

; together they draw a surface above the

x y

-floor.

What does the subscript in

f_{x}

tell you to do?

Measure the slope in the

x

-direction while freezing

y

as a constant.

Compute

f_{x}

for

f = x^{2} + 3 x y

f_{x} = 2 x + 3 y

(treat

y

as a constant).

What does

\nabla f = 0

mean in plain words?

Both slopes

f_{x}

and

f_{y}

are zero — the ground is flat, a critical point.

What is the difference between

f_{xx}

and

f_{x y}

f_{xx}

is curvature along

x

;

f_{x y}

is the twist — how the

x

-slope changes as you step in

y

Write the Hessian

H

and its determinant.

H = (f_{xx} f_{x y} f_{x y} f_{y y})

, and

det H = f_{xx} f_{y y} - (f_{x y})^{2}

Why is the height-change

Q

near a critical point quadratic, not linear?

The linear (slope) part is zero there, so the quadratic curvature part leads.

What does

\approx

signal in the Taylor step?

An approximation valid for tiny steps, where the ignored higher-order terms are negligible.

Connections

Critical points — finding, classifying — the parent topic these foundations build toward.
Gradient and Directional Derivatives — where $\nabla f$ and directional slope come from.
Hessian Matrix — the curvature grid $H$ and its determinant.
Taylor Series in Several Variables — supplies the quadratic form $Q$ .
Quadratic Forms and Definiteness — reads the sign of $Q$ (min/max/saddle).
Single-variable Second Derivative Test — the 1-D warm-up.
Lagrange Multipliers — extrema under constraints.