Critical points — finding, classifying — Hinglish notes

4.4.12 · Maths › Multivariable Calculus

KIYA hai ek critical point?

KYUN sirf critical points par? Ek smooth local extremum par, $x$ -direction mein chalo: tum ek 1-D max/min par ho, isliye $f_{x} = 0$ . $y$ ke liye bhi same. Agar kisi bhi direction mein slope nonzero hoti, toh tum downhill (ya uphill) step le sakte, toh woh extremum nahi hota. Yeh Fermat's theorem ka generalised version hai.

KAISE dhundhen unhe

$f_{x}$ aur $f_{y}$ compute karo.
System $f_{x} = 0, f_{y} = 0$ ko simultaneously solve karo.
Woh points bhi flag karo jahan partials exist nahi karti (corners, cusps).

KAISE classify karein — Second Derivative Test

Test ko Hessian se derive karna

Critical point $(a, b)$ ke paas Taylor-expand karo. $h, k$ small steps hain. Kyunki $f_{x} = f_{y} = 0$ : $f (a + h, b + k) - f (a, b) \approx \frac{1}{2} (f_{xx} h^{2} + 2 f_{x y} hk + f_{y y} k^{2}) .$

Yeh quadratic form sab kuch decide karta hai. Ise Hessian matrix ke saath likho: $H = (f_{xx} f_{x y} f_{x y} f_{y y}), Q (h, k) = \frac{1}{2} (h k) H (h k)^{⊤} .$

Kyun complete the square? $Q$ ka sign sab directions ke liye ek saath dekhne ke liye. Assume karo $f_{xx} \neq = 0$ : $f_{xx} h^{2} + 2 f_{x y} hk + f_{y y} k^{2} = f_{xx} (h + \frac{f _{x y}}{f _{xx}} k)^{2} + (f_{y y} - \frac{f _{x y}^{2}}{f _{xx}}) k^{2} .$

Doosra bracket equals $\frac{f _{xx} f _{y y} - f _{x y}^{2}}{f _{xx}} = \frac{D}{f _{xx}}$ hai, jahan

Ab signs padho:

Agar $D > 0$ aur $f_{xx} > 0$ : dono squared terms ke positive coefficients hain $\Rightarrow Q > 0$ hamesha $\Rightarrow$ local minimum.
Agar $D > 0$ aur $f_{xx} < 0$ : dono coefficients negative $\Rightarrow Q < 0$ hamesha $\Rightarrow$ local maximum.
Agar $D < 0$ : dono square-coefficients ke opposite signs hain $\Rightarrow Q$ kuch directions mein positive, kuch mein negative $\Rightarrow$ saddle point.
Agar $D = 0$ : quadratic degenerate ho jaata hai; test inconclusive hai — higher terms ya direct inspection use karo.

Figure — Critical points — finding, classifying

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall Second derivative test ke chaar outcomes kya hain?

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

Recall Extrema sirf wahan kyun ho sakte hain jahan

$\nabla f = 0$ ho (smooth $f$ ke liye)? Agar koi bhi directional slope nonzero hoti toh tum downhill ya uphill step le sakte, jo max/min hone se contradict karta. Isliye sab partials vanish hone chahiye.

Recall

$D = f_{xx} f_{y y} - f_{x y}^{2}$ kahan se aata hai? Yeh $det H$ hai, woh coefficient jo 2nd-order Taylor expansion mein square complete karne ke baad bachta hai; iska sign decide karta hai ki quadratic form definite hai ya indefinite.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek bumpy blanket socho. Flat spots special hain: ek bump ka top, ek dip ka bottom, ya ek horse-saddle shape jahan woh ek taraf upar aur doosri taraf neeche jaati hai. Decide karne ke liye, feel karo blanket kaise curve karti hai. Har jagah upar curve = bowl (lowest point). Har jagah neeche curve = hill (highest point). Ek taraf upar aur doosri taraf neeche curve = saddle. Ek single number $D$ hamare liye yeh curve-checking karta hai, aur ek doosra number bowl ko hill se alag karta hai.

Definition of a critical point of

f (x, y)

Woh point jahan

\nabla f = 0

(yaani

f_{x} = f_{y} = 0

) ho ya jahan koi partial exist na kare.

Formula for the discriminant

D

D = f_{xx} f_{y y} - (f_{x y})^{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 (

f_{xx}

ki parwah kiye bina).

D = 0

implies what?

Test inconclusive hai; seedha inspect karo ya higher-order terms use karo.

Why does

f_{x y}

matter for classification?

Yeh cross-curvature measure karta hai; bada

∣ f_{x y} ∣

D < 0

kar sakta hai aur ek apparent min/max ko saddle mein badal sakta hai.

Classify

(0, 0)

for

f = x^{2} - y^{2}

Saddle, kyunki

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

What test step comes BEFORE checking

f_{xx}

D

compute karna; sirf agar

D > 0

ho tab

f_{xx}

padhna.

Where does

D

come from in the derivation?

Yeh woh surviving coefficient hai jo 2nd-order Taylor quadratic form mein square complete karne ke baad milta hai (Hessian determinant).

Connections

Gradient and Directional Derivatives — $\nabla f = 0$ critical points define karta hai.
Hessian Matrix — $D = det H$ ; eigenvalue signs test ko generalise karte hain.
Taylor Series in Several Variables — quadratic-form derivation ka source.
Quadratic Forms and Definiteness — positive/negative/indefinite ↔ min/max/saddle.
Lagrange Multipliers — extrema dhundhna constraints ke saath.
Single-variable Second Derivative Test — 1-D special case.

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

Critical points — finding, classifying

KIYA hai ek critical point?

KAISE dhundhen unhe

KAISE classify karein — Second Derivative Test

Test ko Hessian se derive karna

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Connections

Concept Map

Deep Dive