Absolute extrema on closed bounded regions: Foundations

4.4.14 · D1 · Maths › Multivariable Calculus › Closed bounded regions par absolute extrema

Parent topic ka recipe run karne se pehle, aapko use padhna aana chahiye. Neeche har woh symbol aur idea hai jo parent use karta hai, ek aisi order mein jahan har cheez sirf pehle waali cheez par rely karti hai.

0. "Highest" aur "lowest" ka matlab kya hai?

Topic ko yeh kyun chahiye: parent ka goal — "absolute max aur min dhundho" — meaningless hai jab tak hum agree nahi karte ki inका matlab hai poore $D$ par sabse uncha aur sabse chhota point.

1. $f (x, y)$ kya hai? — ek field ke upar height

Yahan picture hi sab kuch hai.

Figure — Absolute extrema on closed bounded regions

Figure s01 — Alt text: A curved surface (terrain) floats above a flat $x$ – $y$ floor. One floor address $(x, y)$ is marked with a navy dot; a vertical navy line rises from it to a magenta dot on the surface labelled $f (x, y)$ , showing that the function value is the height above that address.

Floor woh flat plane hai jis par saare addresses $(x, y)$ hain.
Har address ke upar $f (x, y)$ height par ek dot hai.
Woh saare dots milkar ek surface banate hain — woh "terrain".

Topic ko yeh kyun chahiye: poora game hai "is terrain ka top aur bottom dhundho", toh pehle yeh jaanna zaroori hai ki terrain wahi hai jo $f$ draw karta hai.

2. Region $D$ kya hai? — fenced field

$D$ parent mein do tarahon se aata hai:

Topic ko yeh kyun chahiye: guarantee ("ek max aur min exist karta hai") tabhi true hai kyunki $D$ ek accha, contained patch hai. Fence nahi, guarantee nahi.

3. Distance aur $x^{2} + y^{2}$ — centre se kitni door

Figure s02 — Alt text: A right triangle inside the unit circle. The horizontal leg (length $x$ ) and vertical leg (length $y$ ) meet at a right angle; the dashed hypotenuse from the origin to the point $(x, y)$ is labelled $r = x^{2} + y^{2}$ , illustrating Pythagoras.

$x^{2} + y^{2} < 1$ , i.e. $r < 1$ → radius $1$ wale circle ke strictly andar (open disk).
$x^{2} + y^{2} = 1$ , i.e. $r = 1$ → exactly circle PAR (boundary).
$x^{2} + y^{2} \leq 1$ , i.e. $r \leq 1$ → andar YA us par (closed disk).

Topic ko yeh kyun chahiye: disk region aur "kya $(\frac{1}{2}, \frac{1}{2})$ andar hai?" wala check dono seedha $r^{2} = x^{2} + y^{2}$ ko $1$ se compare karne se aate hain. Example 2 mein, $\frac{1}{4} + \frac{1}{4} = \frac{1}{2} < 1$ , toh $r < 1$ aur point safely andar hai.

4. Interior, boundary, closed, bounded — field ki shape

Figure s03 — Alt text: A shaded disk with a solid violet boundary circle. A magenta dot near the middle is labelled "interior point (wiggle room all ways)"; a navy dot on the boundary circle is labelled "boundary point (on the fence)". A dotted larger circle around everything is labelled "fits in a big circle → bounded".

Figure dekho: pale middle mein ek point interior hai; dark rim par ek point boundary hai. Agar rim solid draw hai, toh set uska malik hai — yahi closed hai.

Topic ko yeh kyun chahiye: parent ki strategy literally "interior" vs "boundary" mein split hoti hai, toh yeh words woh coat-hooks hain jis par poora recipe tika hua hai.

5. Continuous — koi rips ya cliffs nahi

Figure s05 — Alt text: Two side-by-side height slices. On the left, a smooth continuous curve labelled "continuous — small step in x → small change in height". On the right, a curve with a vertical jump (a torn cliff) at one point, an open circle below and a filled circle above, labelled "discontinuous — a tiny step causes a sudden jump"; a magenta arrow points at the gap.

Topic ko yeh kyun chahiye: agar sheet mein rip hoti, toh "highest point" torn edge par chhup sakta tha aur kabhi attain nahi hota. Continuity Extreme Value Theorem ki ek required ingredient hai.

6. Partial derivatives $f_{x}$ aur $f_{y}$ — ek direction mein slope

Figure s04 — Alt text: A height-versus-step graph showing two curves — an orange parabola (the slice with $y$ fixed, whose slope is $f_{x}$ ) and a magenta line (the slice with $x$ fixed, whose slope is $f_{y}$ ). A dashed violet tangent line touches the orange curve at a marked point, labelled "steepness here = $f_{x}$ ".

Derivatives kyun, kuch aur kyun nahi? Ek derivative exactly yeh sawaal answer karta hai — "uphill kis taraf hai, aur kitna steep?" — woh precise sawaal jo hum ek flat spot dhundhne ke liye poochte hain. Yeh machinery Critical points and gradient mein develop ki gayi hai.

7. Gradient $\nabla f$ — poora uphill arrow

Figure s06 — Alt text: A surface with a contour/floor beneath it. At one floor point a violet arrow labelled " $\nabla f$ " points in the steepest-uphill direction; a magenta dashed path shows the surface climbing most steeply above that arrow. At a separate flat dimple the arrow has shrunk to a dot labelled " $\nabla f = 0$ — level, a critical point".

Topic ko yeh kyun chahiye: recipe ka Step 1 literally hai " $D$ ke andar $\nabla f = 0$ solve karo".

8. Boundary ko parametrize karna — fence ko ek single dial mein badalna

Intuition Ek rectangle mein CHAAR edges aur CHAAR corners hain — sabko karo

Rectangle boundary ek curve nahi balki chaar corners par judi chaar straight edges hain. Har edge ko is tarah handle karo: jo coordinate pin hai use fix karo aur doosre ko uske interval mein vary hone do, phir corners ko mandatory extra candidates ki tarah treat karo:

Bottom $y = 0$ , with $0 \leq x \leq 3$ : $f$ akele $x$ ka function ban jaata hai.
Top $y = 2$ , with $0 \leq x \leq 3$ : $f$ akele $x$ ka function ban jaata hai.
Left $x = 0$ , with $0 \leq y \leq 2$ : $f$ akele $y$ ka function ban jaata hai.
Right $x = 3$ , with $0 \leq y \leq 2$ : $f$ akele $y$ ka function ban jaata hai. Har edge par aap uske interior critical points aur uske dono endpoints dono check karte ho. Endpoints woh corners $(0, 0), (3, 0), (0, 2), (3, 2)$ hain — jahan do edges milti hain, aur har ek mandatory candidate hai kyunki ek closed interval par 1-variable function ek endpoint par peak kar sakta hai (dekho Extreme Value Theorem (1D)). Kisi corner ko skip karna true winner ko kho sakta hai: parent ke Example 1 mein absolute max $9$ exactly corner $(3, 0)$ par baitta hai.

Topic ko yeh kyun chahiye: har boundary piece — rectangle ki har edge, disk ka poora rim, aur har joining corner — ko ek aise form mein squeeze karna hoga jahan ordinary 1D methods apply hon.

Prerequisite map

Neeche diya diagram ek dependency chart hai: ek arrow "A → B" ka matlab hai "B samajhne se pehle A chahiye". Ise top-to-bottom padho. Teen raw ingredients — terrain $f$ , Pythagoras (distance ke liye), aur continuity — shape-words aur derivatives mein feed hote hain; woh baad mein do recipe steps ko power dete hain, jo finally milkar parent recipe bante hain. Neeche wali figure usi structure ko ek picture ke roop mein render karti hai taaki aap flow dekh sako, sirf padho nahi.

Figure s07 — Alt text: A top-down dependency flowchart. Boxes for "function f", "Pythagoras (x²+y²)", and "continuous" sit at the top; arrows lead down through "region D", "interior/boundary/closed/bounded", "partial derivatives", and "gradient ∇f" into two boxes "Step 1 interior critical points" and "Step 2 boundary as one variable", which both feed the bottom box "Absolute extrema recipe". "Extreme Value Theorem 2D" also feeds the recipe.

Equipment checklist

Ek point global maximum kyun hota hai (sirf "high" nahi)?

Uski height

D

ke har doosre point ki height se

\geq

hoti hai — poore FIELD par sabse uncha, sirf nearby bumps se uncha nahi.

f (x, y)

ek picture ki tarah kya represent karta hai?

Har floor address

(x, y)

ke upar float karta hua ek height/terrain.

{0 \leq x \leq 3, 0 \leq y \leq 2}

kya describe karta hai?

Un saare points ka set jahan

x

0 aur 3 ke beech ho AUR

y

0 aur 2 ke beech ho — ek rectangle.

r

formula ke roop mein kya hai, aur yeh kabhi negative kyun nahi hota?

r = x^{2} + y^{2}

, origin se distance; ek distance ek length hai, toh square root apni non-negative value leti hai.

x^{2} + y^{2}

tab bhi kyun kaam karta hai jab

x

y

negative ho?

Squaring sign mita deti hai, toh yeh chaaon quadrants mein correct distance-squared deta hai.

x^{2} + y^{2} < 1

= 1

, aur

\leq 1

mein difference kya hai?

r < 1

strictly andar;

r = 1

exactly boundary par;

r \leq 1

andar ya us par (closed disk).

"Closed" aur "bounded" ka matlab kya hai?

Closed = apni boundary fence include karta hai (

\leq

); bounded = kisi finite circle ke andar fit ho jaata hai.

Guarantee ke liye

f

continuous kyun hona chahiye?

Ek rip height ko kisi value ki taraf approach karne deta hai bina actually reach kiye, toh koi genuine highest point exist nahi karta.

f_{x}

kya hai aur kya fixed rakha jaata hai?

x

-direction mein chalte hue slope,

y

ko constant rakhte hue.

\nabla f

kya hai aur

\nabla f = 0

geometrically kya mean karta hai?

Arrow

(f_{x}, f_{y})

jo sabse steep uphill point karta hai; zero ka matlab ground har direction mein level hai (ek critical point).

Interior max sirf

\nabla f = 0

par hi kyun ho sakta hai?

Agar arrow nonzero hota toh aap uphill step le sakte the aur upar ja sakte the, toh woh max tha hi nahi.

Rectangle boundary ke chaaon edges aur corners ko kaise handle karte hain?

Har edge par pinned coordinate fix karo (

f

ko 1-variable function banao), uske interior critical points dhundho, aur chaaon corner endpoints ko hamesha candidates ki tarah include karo.

Unit circle boundary ko ek variable mein kaise badle?

x = cos t, y = sin t

set karo aur angle

t

ko single dial banao.