Critical points — finding, classifying: Foundations

4.4.12 · D1 · Maths › Multivariable Calculus › Critical points — finding, classifying

Is page par kuch bhi assume nahi kiya gaya. Agar parent note mein koi symbol use hua tha aur expect kiya tha ki tum pehle se jaante ho, toh hum use yahan pehle ek picture se build karte hain. Upar se neeche padho — har idea woh zameen hai jis par agla idea khada hota hai.

0. Woh picture jisme sab kuch hota hai

Kisi bhi symbol se pehle, mental image fix karo. Do variables ka function $f (x, y)$ ek height machine hai: tum use do numbers dete ho (flat map par ek jagah, jaise east-position $x$ aur north-position $y$ ) aur woh tumhe ek number wapas deta hai — us jagah ki zameen ki height. Us height ko hum $z$ kehte hain.

Figure — Critical points — finding, classifying

Yeh topic ko kyun chahiye: critical points woh khaas floor par spots $(x, y)$ hain jahan upar ki surface kuch interesting karti hai (peak, pit, pass). Surface nahi toh kahani nahi.

1. Coordinates $x, y$ aur point $(a, b)$

Dono inputs ke naam $x$ aur $y$ hain. Koi khaas spot jis par hum dhyan dete hain usse aksar apne letters milte hain, usually $(a, b)$ — socho jaise "humne east-position $a$ , north-position $b$ par ek flag lagaya."

Kyun: parent likhta hai jaise " $(a, b)$ ek critical point hai" — yeh ek dot ke baare mein haan/na ka sawaal pooch raha hai.

2. Ek direction mein slope: partial derivative $f_{x}$

Surface par khade ho. Ab sirf east ki taraf chalo ( $x$ badhao, $y$ frozen rakho). Tum ek single curve trace karte ho — surface ka ek slice. Us slice ki steepness ek slope hai. $y$ ko freeze karke east-west slope measure karna hume partial derivative with respect to $x$ deta hai, jise $f_{x}$ likha jaata hai.

Yeh topic ko kyun chahiye: ek flat spot east–west AUR north–south dono mein flat hona chahiye. Yahi exactly do conditions hain $f_{x} = 0$ aur $f_{y} = 0$ . Poori directional story ke liye Gradient and Directional Derivatives dekho.

3. Dono slopes ko bundle karna: gradient $\nabla f$

Do slopes $f_{x}$ aur $f_{y}$ zameen ko poori tarah describe karte hain. Hum inhe ek arrow mein staple karte hain jise gradient kehte hain, likha jaata hai $\nabla f$ (symbol $\nabla$ ko "nabla" ya "del" padha jaata hai — bas "partial slopes collect karo" ka naam).

Kyun: parent ki critical point ki poori definition ek line hai — $\nabla f (a, b) = 0$ . Ab tum ise padh sakte ho: "sabse steep uphill arrow sikar ho gaya, kyunki koi uphill nahi — tum ek flat spot par ho."

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

Flat spot par arrow gone hai, toh slope hume shape ke baare mein kuch nahi batata. Hume chahiye slope khud kitna change ho raha hai — yahi curvature hai. Ek derivative ka derivative lo.

$f_{xx}$ = "east-slope kitna change hota hai jab tum east mein chalna jaari rakhte ho" = $x$ ke saath upar-curve ya neeche-curve. Positive = bowl-shaped (cup 🙂), negative = dome-shaped (cap 🙁).
$f_{y y}$ = north direction mein same idea.
$f_{x y}$ = mixed wala: "east-slope kitna change hota hai jab tum north mein kadam rakhte ho?" Yeh measure karta hai ki dono directions kitni ek doosre ke saath twist karti hain.

Yeh topic ko kyun chahiye: yeh teen numbers Hessian Matrix aur poore Second Derivative Test ka raw material hain.

5. Curvature ko package karna: Hessian matrix $H$ aur $det H$

Teen curvature numbers ek saaf box deserve karte hain. Matrix bas numbers ka ek rectangular grid hota hai. Hessian $H$ second partials ka 2×2 grid hai.

Kyun: parent ka boxed formula $D = det H$ ab magic nahi hai — yeh hai "seedhi curvatures multiply karo, twist squared ghata do."

6. Quadratic ka sign padhna: quadratic form $Q$

Flat spot ke paas, ek chote kadam $(h, k)$ ke liye height mein badlaav (approximately) ek quadratic form hota hai — squared-ish terms ka sum: $Q (h, k) = \frac{1}{2} (f_{xx} h^{2} + 2 f_{x y} hk + f_{y y} k^{2}) .$

Yahan $h, k$ bas east aur north mein chote step sizes hain. "Quadratic" ka matlab hai har term degree-2 ka hai (har variable do baar aata hai: $h^{2}$ , $hk$ , $k^{2}$ ).

Kyun: yeh quadratic bridge hai — Taylor Series in Several Variables se banaya hua — raw curvature numbers aur plain-English verdict ke beech.

7. Woh tool jo $Q$ ko appear karta hai: Taylor expansion aur " $\approx$ "

Hume pata kaise chalta hai ki flat spot ke paas height change quadratic dikhti hai? Kyunki Taylor expansion kisi bhi smooth surface ko ek point ke paas ek polynomial se approximate karta hai: constant + linear + quadratic + chhoti corrections.

Ek critical point par constant part bas height khud hai aur linear part zero hai (slopes gone), jo $Q$ ko leading actor banata hai. Is poore idea ka 1-D warm-up Single-variable Second Derivative Test hai; constraints ko alag se Lagrange Multipliers handle karta hai.

Yeh topic ko kaise feed karte hain

Equipment checklist

Left side padho, jawab do, phir reveal karo.

f (x, y)

kya output karta hai, aur uske saare outputs kaisi shape banate hain?

Ek height number

z

; saath mein woh

x y

-floor ke upar ek surface draw karte hain.

f_{x}

mein subscript tumhe kya karne ko kehta hai?

x

-direction mein slope measure karo jabki

y

ko constant freeze karo.

f = x^{2} + 3 x y

ke liye

f_{x}

compute karo.

f_{x} = 2 x + 3 y

(

y

ko constant maano).

\nabla f = 0

ka plain words mein kya matlab hai?

Dono slopes

f_{x}

aur

f_{y}

zero hain — zameen flat hai, ek critical point.

f_{xx}

aur

f_{x y}

mein kya fark hai?

f_{xx}

x

ke saath curvature hai;

f_{x y}

twist hai —

x

-slope kitna change hota hai jab tum

y

mein kadam rakhte ho.

Hessian

H

aur uska determinant likho.

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

, aur

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

Critical point ke paas height-change

Q

quadratic kyun hai, linear kyun nahi?

Linear (slope) part wahan zero hai, toh quadratic curvature part lead karta hai.

Taylor step mein

\approx

kya signal karta hai?

Ek approximation jo chote steps ke liye valid hai, jahan ignore kiye gaye higher-order terms negligible hain.

Connections

Critical points — finding, classifying — woh parent topic jis ke liye yeh foundations build ho rahi hain.
Gradient and Directional Derivatives — $\nabla f$ aur directional slope kahan se aate hain.
Hessian Matrix — curvature grid $H$ aur uska determinant.
Taylor Series in Several Variables — quadratic form $Q$ supply karta hai.
Quadratic Forms and Definiteness — $Q$ ka sign padhta hai (min/max/saddle).
Single-variable Second Derivative Test — 1-D warm-up.
Lagrange Multipliers — constraints ke under extrema.