Second derivative test — Hessian determinant: Foundations

4.4.13 · D1 · Maths › Multivariable Calculus › Second derivative test — Hessian determinant

Is page ko padh ke laga ki tumhe kuch nahi pata. Second derivative test — Hessian determinant padhne se pehle tumhe symbols aur pictures ki ek chhoti toolkit chahiye. Hum har ek cheez zero se banate hain, ek aisi sequence mein jahan har naya idea sirf pehle waale ideas par tikta hai.

1. $f (x, y)$ kya hai? — ek landscape

Socho ek pahadi ilake ka map. Map par har dot ka ek east–west coordinate $x$ hai aur north–south coordinate $y$ hai. Function $f$ tumhe wahan ki altitude batata hai. Un sab altitudes ko stack karo aur tumhe ek surface milega jo flat $x y$ -plane ke upar tairti hai.

Us vertical altitude ko hum apna naam dete hain. $z$ ko map ke floor se kitne upar ho — upar–neeche ka coordinate, jo $x$ aur $y$ dono se right angles par hai — ke liye use karo. Tab surface sirf un sab points ka collection hai jahan height wahi hai jo $f$ return karta hai:

$z = f (x, y)$

Figure — Second derivative test — Hessian determinant

Flat floor $x y$ -plane hai — map, upar se dekha hua.
Vertical coordinate $z$ altitude measure karta hai; equation $z = f (x, y)$ har map-point ko surface (red) tak utha deta hai.
Yeh kyun chahiye: baaki sab cheez — slopes, curving, critical points — is surface ke baare mein hi kahi gayi baatein hain.

2. Partial derivative $f_{x}$ kya hai? — EK direction mein slope

Pahad par tum kai directions mein chal sakte ho aur steepness alag-alag hoti hai. Partial derivative yeh jawab deta hai: agar main bilkul ek compass direction mein chaloon, toh kitna steep hai?

Chhota subscript woh direction hai jiske against baaki sab freeze kiya. $f_{x}$ compute karne ke liye $y$ ko plain constant maan ke differentiate karo, aur vice versa.

$f_{x} > 0$ : east kadam rakhte hi zameen upar jaati hai (red arrow uphill point karta hai).
$f_{x} < 0$ : east kadam rakhte hi zameen neeche jaati hai.
$f_{x} = 0$ : zameen east–west direction mein momentarily flat hai.

Topic ko yeh kyun chahiye: test ka pehla step yahi hai ki dhundhna jahan zameen har direction mein flat ho — aur "har direction" sirf $f_{x} = 0$ aur $f_{y} = 0$ dono saath mein capture ho jaata hai.

3. Gradient $\nabla f$ kya hai, aur critical point kya hota hai?

Agar ek bhi slope nonzero hoti, tum abhi bhi uphill ya downhill kisi taraf chal sakte, toh tum peak, pit, ya pass par nahi ho sakte. Sirf jab dono zero ho jaate hain tab tangent plane horizontal hoti hai.

Topic ko yeh kyun chahiye: poora test sirf critical points par hi valid hai. $\nabla f = 0$ solve karne ke liye dekho Critical points and gradient.

4. Second derivatives — slope nahi, CURVING

Flat spot par slope har direction mein zero hai, phir bhi woh spot pit, peak, ya pass ho sakta hai. Inhe alag karta hai slope nahi balki slope kitna change ho raha hai — curving.

Subscripts left to right padho jaise "pehle is se differentiate karo, phir us se." Kyunki critical point par slope zero tha, curving pehli cheez hai jo real information carry karti hai — yahi wajah hai test ko second derivative test kehte hain.

5. Clairaut's theorem — dono mixed partials agree karte hain

"East phir north" differentiate karna wahi answer deta hai jaise "north phir east." Toh chaar second derivatives actually sirf teen distinct numbers hain: $f_{xx}$ , $f_{y y}$ , aur shared cross-term $f_{x y}$ . Yahi wajah hai ki agli step mein matrix symmetric hai (diagonal ke across ek mirror). Dekho Clairaut's theorem.

6. Hessian matrix mein pack karna

Inhe bundle kyun karein? Kyunki surface ki shape teeno curvatures se ek saath control hoti hai, aur matrix "curving jo direction par depend kare" ke liye natural container hai. Dekho Hessian matrix.

Woh last number $D$ parent note ka star hai. Subtract kiya gaya $(f_{x y})^{2}$ exactly step 4 ka tilt term hai — yeh optional nahi hai.

7. Quadratic form $Q (u, v)$ — local shape

Critical point ke paas, agar hum $u = x - a$ aur $v = y - b$ ko chhote steps maanein, surface ki local shape ek akele expression se capture hoti hai:

Woh tool jo yeh expression produce karta hai — constant aur (zero) linear terms drop karke second-order part rakhna — multivariable Taylor expansion hai. Dekho Taylor series multivariable. Aur jab aisa expression ek hi sign rakhe us ka study exactly Quadratic forms and definiteness hai.

Yahan kyun chahiye: yeh bridge hai. Slopes ne bataya kahan dekhna hai (critical point); quadratic form batata hai kaisi shape hai wahan, aur $D$ woh shape read kar leta hai.

Prerequisite map

Equipment checklist

Recall Apni readiness self-test karo

$f (x, y)$ kya return karta hai, aur iske saath kaun sa picture jaata hai? ::: Ek number = surface ki height map-point $(x, y)$ ke upar. Coordinate $z$ kiske liye hai? ::: $x y$ -plane ke upar vertical altitude; surface hai $z = f (x, y)$ . $f_{x}$ geometrically kya measure karta hai? ::: Surface ka slope agar tum sirf $+ x$ direction mein chalo, $y$ ko freeze karte hue. Gradient $\nabla f$ kya hai? ::: Pair $(f_{x}, f_{y})$ ; map par ek arrow jo steepest-uphill point karta hai. $(a, b)$ critical point kab hai? ::: Jab $f_{x} (a, b) = 0$ aur $f_{y} (a, b) = 0$ — dono directions mein flat, isliye sab mein. $f_{xx}$ kya measure karta hai, $f_{x}$ se alag? ::: $f_{xx}$ curving hai (slope kitna change hota hai), slope khud nahi. Mixed partial $f_{x y}$ kya represent karta hai? ::: Dono directions ke beech tilt/coupling. Hessian symmetric kyun hai? ::: Clairaut's theorem: $f_{x y} = f_{y x}$ jab $f$ $C^{2}$ ho (second partials continuous). $det (a c b d)$ compute karo. ::: $a d - b c$ . Hessian ke liye $D$ likho. ::: $D = f_{xx} f_{y y} - (f_{x y})^{2}$ . Quadratic form $Q (u, v)$ kya hai? ::: $f_{xx} u^{2} + 2 f_{x y} uv + f_{y y} v^{2}$ — critical point ke paas local shape.

Connections

Second derivative test — Hessian determinant — woh parent jisme yeh page feed karta hai.
Critical points and gradient — $\nabla f = 0$ solve karna.
Taylor series multivariable — quadratic form kahan se aata hai.
Hessian matrix — matrix aur uske eigenvalues.
Quadratic forms and definiteness — jab $Q$ ek sign rakhe.
Clairaut's theorem — kyun $f_{x y} = f_{y x}$ .
Lagrange multipliers — constrained cousin.