Critical points — finding, classifying

WHAT is a critical point?

WHY only at critical points? At a smooth local extremum, walk along the $x$ -direction: you're at a 1-D max/min, so $f_x=0$ . Same for $y$ . If the slope in any direction were nonzero, you could step downhill (or uphill), so it wasn't an extremum. This is Fermat's theorem generalised.

HOW to find them

Compute $f_x$ and $f_y$ .
Solve the system $f_x=0,\ f_y=0$ simultaneously.
Also flag points where partials don't exist (corners, cusps).

HOW to classify — the Second Derivative Test

Deriving the test from the Hessian

Taylor-expand near a critical point $(a,b)$ . Let $h,k$ be small steps. Since $f_x=f_y=0$ : $f(a+h,b+k) - f(a,b) \approx \tfrac12\big(f_{xx}h^2 + 2f_{xy}hk + f_{yy}k^2\big).$

This quadratic form decides everything. Write it with the Hessian matrix: $H=\begin{pmatrix} f_{xx} & f_{xy}\\ f_{xy} & f_{yy}\end{pmatrix}, \qquad Q(h,k)=\tfrac12\,(h\ k)\,H\,(h\ k)^{\!\top}.$

Why complete the square? To see the sign of $Q$ for all directions at once. Assume $f_{xx}\neq0$ : $f_{xx}h^2 + 2f_{xy}hk + f_{yy}k^2 = f_{xx}\Big(h+\tfrac{f_{xy}}{f_{xx}}k\Big)^2 + \Big(f_{yy}-\tfrac{f_{xy}^2}{f_{xx}}\Big)k^2.$

The second bracket equals $\dfrac{f_{xx}f_{yy}-f_{xy}^2}{f_{xx}} = \dfrac{D}{f_{xx}}$ , where

Now read the signs:

If $D>0$ and $f_{xx}>0$ : both squared terms have positive coefficients $\Rightarrow Q>0$ always $\Rightarrow$ local minimum.
If $D>0$ and $f_{xx}<0$ : both coefficients negative $\Rightarrow Q<0$ always $\Rightarrow$ local maximum.
If $D<0$ : the two square-coefficients have opposite signs $\Rightarrow Q$ is positive some directions, negative others $\Rightarrow$ saddle point.
If $D=0$ : the quadratic degenerates; test is inconclusive — use higher terms or direct inspection.

Figure — Critical points — finding, classifying

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall What are the four outcomes of the second derivative test?

$D>0,f_{xx}>0$ : min. $D>0,f_{xx}<0$ : max. $D<0$ : saddle. $D=0$ : inconclusive.

Recall Why can extrema only occur where

$\nabla f=0$ (for smooth $f$ )? If any directional slope were nonzero you could step downhill or uphill, contradicting being a max/min. Hence all partials must vanish.

Recall Where does

$D=f_{xx}f_{yy}-f_{xy}^2$ come from? It's $\det H$ , the leftover coefficient after completing the square in the second-order Taylor expansion; its sign decides whether the quadratic form is definite or indefinite.

Recall Feynman: explain to a 12-year-old

Picture a bumpy blanket. Flat spots are special: top of a bump, bottom of a dip, or a horse-saddle shape where it goes up one way and down another. To tell which, feel how the blanket curves. Curves up everywhere = bowl (lowest point). Curves down everywhere = hill (highest point). Curves up one way and down the other = saddle. A single number $D$ does this curve-checking for us, and a second number tells bowl from hill.

Definition of a critical point of

f(x,y)

A point where

\nabla f=\mathbf 0

(i.e.

f_x=f_y=0

) or where a partial fails to exist.

Formula for the discriminant

D

D=f_{xx}f_{yy}-(f_{xy})^2=\det H

D>0

and

f_{xx}>0

implies what?

Local minimum.

D>0

and

f_{xx}<0

implies what?

Local maximum.

D<0

implies what?

Saddle point (regardless of

f_{xx}

D=0

implies what?

Test is inconclusive; inspect directly or use higher-order terms.

Why does

f_{xy}

matter for classification?

It measures cross-curvature; large

|f_{xy}|

can make

D<0

turning an apparent min/max into a saddle.

Classify

(0,0)

for

f=x^2-y^2

Saddle, since

D=(2)(-2)-0=-4<0

What test step comes BEFORE checking

f_{xx}

Computing

D

; only if

D>0

do you read

f_{xx}

Where does

D

come from in the derivation?

It is the surviving coefficient after completing the square in the 2nd-order Taylor quadratic form (the Hessian determinant).

Connections

Gradient and Directional Derivatives — $\nabla f=0$ defines critical points.
Hessian Matrix — $D=\det H$ ; eigenvalue signs generalise the test.
Taylor Series in Several Variables — source of the quadratic-form derivation.
Quadratic Forms and Definiteness — positive/negative/indefinite ↔ min/max/saddle.
Lagrange Multipliers — finding extrema with constraints.
Single-variable Second Derivative Test — the 1-D special case.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Socho tum ek pahaadi zameen pe khade ho jiska height $z=f(x,y)$ hai. Critical point woh jagah hai jahan zameen har direction mein flat lagti hai — yaani gradient zero, $f_x=0$ aur $f_y=0$ . Yeh teen tarah ke ho sakte hain: gaddha (minimum), choti (maximum), ya ghode ki kaathi yaani saddle, jahan ek taraf upar jaata hai aur doosri taraf neeche.

Pehla step: dono partial derivatives nikaalo aur unhe zero karke equations solve karo — yeh tumhe critical points de dega. Doosra step: classify karna. Iske liye second derivative test use karte hain. Number nikaalo $D=f_{xx}f_{yy}-(f_{xy})^2$ . Yaad rakho mantra: "D decides, phir $f_{xx}$ describes." Agar $D>0$ aur $f_{xx}>0$ to minimum (bowl, smiley), $D>0$ aur $f_{xx}<0$ to maximum (cap, frown), $D<0$ to seedha saddle (yahan $f_{xx}$ dekhne ki zaroorat hi nahi), aur $D=0$ ho to test fail — direct dekhna padega.

Sabse badi galti jo log karte hain: woh sirf $f_{xx}$ aur $f_{yy}$ ka sign dekh ke min/max bol dete hain, aur mixed term $f_{xy}$ ko bhool jaate hain. Lekin $f_{xy}$ cross-curvature batata hai — agar woh bada hua to $D$ negative ho jaata hai aur cheez actually saddle hoti hai. Isliye hamesha pehle $D$ calculate karo. Yeh $D$ kahaan se aaya? Taylor expansion mein second order terms ka square complete karne pe jo bachta hai woh hi $\det H$ hai — isiliye uska sign hi shape decide karta hai. Maths exams aur optimization (machine learning, economics) dono mein yeh test super important hai.

$D=\det H$	$f_{xx}$	Conclusion
$>0$	$>0$	local minimum
$>0$	$<0$	local maximum
$<0$	any	saddle
$=0$	—	inconclusive

Critical points — finding, classifying

WHAT is a critical point?

HOW to find them

HOW to classify — the Second Derivative Test

Deriving the test from the Hessian

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Multivariable Calculus

Connections