Optimization — constrained, unconstrained, real-world problems

4.1.31 · Maths › Calculus I — Limits & Derivatives

KYUN vanish hoti hai derivative ek extremum par?

Classification test ko scratch se derive karna

Hum ek peak aur valley mein fark karna chahte hain. Critical point $c$ ke paas Taylor expansion use karo (jahan $f^{'} (c) = 0$ ):

$f (c + h) = f (c) + f^{'} (c) h + \frac{1}{2} f^{''} (c) h^{2} + \dots$

Kyunki $f^{'} (c) = 0$ hai, leading change yeh hai:

$f (c + h) - f (c) \approx \frac{1}{2} f^{''} (c) h^{2} .$

Yeh step kyun? $h^{2} > 0$ kisi bhi chote $h \neq = 0$ ke liye, isliye $f^{''} (c)$ ka sign akela decide karta hai ki $f$ dono sides par upar jaata hai ya neeche.

Unconstrained vs Constrained — FARK kya hai?

Lagrange multipliers kyun? (geometric derivation)

Ek constrained optimum par hum curve $g = 0$ ke along chalte hain. Hum $f$ tabhi nahi badha sakte jab $f$ curve ke along badalna band kar de — yaani $\nabla f$ mein $g = 0$ ke tangent mein koi component nahi hota. Iska matlab $\nabla f$ , $\nabla g$ ke parallel hai:

$\nabla f = λ \nabla g, g (x, y) = 0.$

Yeh step kyun? $\nabla g$ constraint curve ke perpendicular hota hai; agar $\nabla f$ bhi perpendicular hai, toh koi bhi tangent direction $f$ ko improve nahi kar sakta. Scalar $λ$ measure karta hai ki agar constraint thoda dhila ho toh optimum kitna badlega.

Universal recipe (80/20 core)

Figure — Optimization — constrained, unconstrained, real-world problems

Worked Example 1 — Maximum volume ka box (constrained → 1-D)

Ek 12 cm × 12 cm ki sheet hai jisme har corner se $x$ side ke equal squares kaate jaate hain; flaps fold hokar ek open box banaate hain. Volume maximize karo.

Fold karne ke baad dimensions: base $(12 - 2 x)$ , height $x$ .
$V (x) = x (12 - 2 x)^{2}$ , valid hai $0 \leq x \leq 6$ ke liye. Kyun? $x < 0$ meaningless hai, $x > 6$ se negative base milta hai.

Differentiate karo (product rule): $V^{'} (x) = (12 - 2 x)^{2} + x \cdot 2 (12 - 2 x) (- 2) = (12 - 2 x) [(12 - 2 x) - 4 x] .$ Factor kyun? Yeh turant roots reveal karta hai. $V^{'} (x) = (12 - 2 x) (12 - 6 x) = 0 \Rightarrow x = 6 or x = 2.$

Endpoints se classify karo: $V (0) = 0, V (6) = 0, V (2) = 2 (8)^{2} = 128$ . Toh $x = 2$ se maximum volume $128 cm^{3}$ milta hai.

Worked Example 2 — Closest point (substitution)

Line $y = 2 x + 1$ par origin ke sabse paas wala point dhundo.

Distance minimize karo, lekin $D$ ki jagah $D^{2}$ minimize karo. Kyun? $\cdot$ increasing hai, isliye minimizer same hoga aur algebra cleaner hogi. $S (x) = x^{2} + y^{2} = x^{2} + (2 x + 1)^{2} = 5 x^{2} + 4 x + 1.$ $S^{'} (x) = 10 x + 4 = 0 \Rightarrow x = - \frac{2}{5} .$ $S^{''} (x) = 10 > 0$ → minimum. Phir $y = 2 (- \frac{2}{5}) + 1 = \frac{1}{5}$ .

Nearest point: $(- \frac{2}{5}, \frac{1}{5})$ , distance $= S = \frac{1}{5} = \frac{1}{5}$ . Sanity check: is point tak ka segment line ke perpendicular hona chahiye (slope $- \frac{1}{2}$ vs line slope $2$ , product $= - 1$ ✓).

Worked Example 3 — Lagrange multipliers

$f (x, y) = x y$ ko $x + y = 10$ subject maximize karo.

$g = x + y - 10$ ke saath $\nabla f = λ \nabla g$ set karo: $y = λ, x = λ \Rightarrow x = y .$ Kyun? $g$ ke dono partials $1$ hain, isliye $f_{x} = f_{y} \Rightarrow y = x$ . Constraint se $2 x = 10 \Rightarrow x = y = 5$ , toh max product $= 25$ .

Yeh famous result recover karta hai: fixed sum wale numbers mein, product tab sabse bada hota hai jab woh equal hoon (AM–GM in disguise).

Recall Feynman: 12-saal ke bachche ko explain karo

Ek roller-coaster track socho. Ek hill ke bilkul top par track ek pal ke liye flat hota hai — woh flatness hi hai "slope equals zero." Toh sabse uuche ya sabse neeche wale spots dhundhne ke liye, woh jagah dhundo jahan track flat ho, phir dekho ki woh hilltop hai ya dip. Agar tum ek path par stuck ho (ek fence jiske along tum chalna padega), tab tum tabhi aur upar nahi chadh sakte jab "upar" wali direction seedha fence ke aarpaar point kare — yahi Lagrange ka idea hai.

Flashcards

Ek differentiable function ke interior extremum par

f^{'} (c)

ko kya satisfy karna chahiye?

f^{'} (c) = 0

(Fermat's Theorem).

Kya har critical point ek extremum hota hai?

Nahi — jaise

x^{3}

mein

0

par inflection hai; critical points sirf candidates hote hain.

Second-derivative test:

f^{'} (c) = 0, f^{''} (c) > 0

ka matlab?

Local minimum.

Second-derivative test:

f^{''} (c) < 0

ka matlab?

Local maximum.

Jab 2nd-derivative test fail ho (

f^{''} = 0

), tab kya use karte hain?

First-derivative sign test.

Distance problems mein

D

ki jagah

D^{2}

minimize kyun karte hain?

\cdot

increasing hai, isliye minimizers same hote hain aur algebra simpler hoti hai.

f

g = 0

s.t. optimize karne ki Lagrange condition kya hai?

\nabla f = λ \nabla g

with

g = 0

\nabla f = λ \nabla g

ka geometric matlab kya hai?

\nabla f

constraint curve ke perpendicular hai, isliye koi bhi tangent move

f

ko improve nahi karta.

Open-box problem mein

x

ko kaun sa domain restrict karta hai?

0 \leq x \leq 6

(base

12 - 2 x \geq 0

Fixed sum

S

wale do numbers ka max product kab hota hai?

Jab woh equal hon, har ek

S /2

[a, b]

par global extremum ke liye kin teen categories ke points compare karne chahiye?

Critical points, aur dono endpoints.

Multiplier

λ

physically kya measure karta hai?

Constraint ko relax karne par optimum ki sensitivity.

Connections

Critical Points & Fermat's Theorem
Second Derivative & Concavity
Taylor Series Expansion
Closed Interval Method (Global Extrema)
Lagrange Multipliers (Multivariable)
AM-GM Inequality
Related Rates — dono words ko derivative equations mein translate karte hain.