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

4.1.31 · D1 · Maths › Calculus I — Limits & Derivatives › Optimization — constrained, unconstrained, real-world proble

Yeh page yeh assume karta hai ki aapne parent note mein use ki gayi koi bhi notation nahi dekhi hai. Hum har symbol ko pehle build karte hain, tabhi usse use karte hain. Upar se neeche padho — har idea sirf pichle wale par tikaa hai.

1. Ek function $f(x)$ — woh machine jo numbers ko numbers mein badaltī hai

Letter $f$ sirf rule ka naam hai (jaise kisi recipe ka naam rakhna). Letter $x$ ek placeholder hai jiske liye bhi hum input daalen. Agar rule hai "ise square karo aur ek joodo," toh $f(x)=x^2+1$, isliye $f(3)=10$.

Topic ko yeh kyun chahiye. Optimization hamesha poochhta hai "kuch cheez ko jitna ho sake bada ya chhota banao." Woh cheez — volume, distance, profit — ek aisa number hai jo kisi choice par depend karta hai. Function exactly yahi hai — "ek number jo ek choice par depend karta hai."

Picture dekho: horizontal axis input $x$ le jaati hai, vertical axis output $f(x)$ le jaati hai. Curve un sab points $(x,f(x))$ ka set hai — har input ko uske output ke saath pair karne ki ek tasveer.

2. Maximum aur minimum — sabse uncha aur sabse neecha point

Ek pahadi landscape ki kalpana karo jo curve ke roop mein bani ho. Teri aankhein pahaadon ke tops (maxima) aur dips ke bottoms (minima) ko turant dhundh leti hain. Optimization woh algebra hai jo unhe graph ko ghoorke dekhe bina dhundh leta hai.

• Local ka matlab hai "sirf paas ke points ke muqaable mein sabse uncha/neecha" — ek pahadi ka top.
• Global ka matlab hai "poore allowed range mein sabse uncha/neecha" — kahin bhi sabse unchi pahadi.

Topic ko yeh kyun chahiye. "Volume maximize karo" literally hai "volume function ka global maximum dhundho." Peaks aur valleys ko naam dena goal ko naam dena hai.

3. Slope — curve kitna steep hai, abhi is waqt

Kisi curve mein zoom karo jab tak apni unglee ke neeche ka tukda ek straight line jaisa nahi lagta. Slope us chhoti straight line ka $\frac{\text{rise}}{\text{run}}$ hai: tum kitna upar jaate ho divided by kitna across.

Figure mein, pale-yellow line ek point par curve ko sirf chhooti hai — yeh tangent line hai, woh straight line jo curve ki direction se wahin match karti hai. Iska steepness hi us point par slope hai.

4. Derivative $f^{\prime }(x)$ — slope apne aap mein ek function ban jaata hai

Chhota mark ${}^{\prime }$ "prime" kehlaata hai. Iska matlab subtraction ya koi naya variable nahi hai — yeh ek label hai jo kehta hai "$f$ se bani slope-machine."

Hume ek poora naya function kyun chahiye, sirf ek number nahi. Ek curve ka slope badalta rehta hai jaise tum uske saath chalte ho — yahan steep, wahin flat, baad mein downhill. Toh slope apne aap mein kuch aisa hai jo depend karta hai ki tum kahaan ho, yaani $x$ ka ek function. Exactly yahi $f^{\prime }(x)$ record karta hai.

Topic ko yeh kyun chahiye — aur yahi tool kyun, koi aur kyun nahi. Hum hazaron $x$-values plug in karke heights compare karne ki koshish kar sakte hain, lekin yeh kabhi prove nahi karta ki humne sach mein peak dhundh li. Derivative ek sharper sawaal ka jawaab deta hai — "curve kahaan rise ya fall karna band karta hai?" — ek single equation $f^{\prime }(x)=0$ se. Isliye calculus, guesswork nahi, optimization ko power karta hai.

Figure mein do graphs stack hain: upar, curve $f$ apni hill aur valley ke saath; neeche, uska derivative $f^{\prime }$. Note karo jahan upar wala curve flat hai, neeche wala curve zero cross karta hai (dashed line). Woh crossing ek extremum ka mechanical fingerprint hai.

5. Second derivative $f^{\prime \prime }(x)$ — slope ka bhi slope

Socho $f^{\prime }$ ko "height kitni tezi se badal rahi hai" aur $f^{\prime \prime }$ ko "slope kitni tezi se badal raha hai." (Agar height position hoti, $f^{\prime }$ velocity hogi aur $f^{\prime \prime }$ acceleration.)

Topic ko yeh kyun chahiye. Ek flat point ($f^{\prime }=0$) akela peak ko valley se alag nahi batata aur na hi kisi aur cheez se. Bending batata hai. Toh $f^{\prime \prime }$ woh tool hai jo ek flat point ko classify karta hai jab hum use dhundh lete hain. Dekho Second Derivative & Concavity.

6. Critical point — jahan flat signal fire karta hai

Letter $c$ sirf "humne jo special input dhundha" uska naam hai, har baar reuse hota hai.

Topic ko yeh kyun chahiye. Critical points candidates hain. Poora game ban jaata hai: critical points list karo, phir har ek ko classify karo. Yeh hai Critical Points & Fermat's Theorem.

7. Endpoints — allowed range ke kinare

Bahut se real problems sirf $x$ ko interval $[a,b]$ ke andar allow karti hain (padho "$a$ se $b$ tak, endpoints included"). Sabse badi ya sabse chhoti value kisi kinare par bhi ho sakti hai chahe curve wahin flat na ho — jaise ek road ka sabse uncha point on-ramp ka top hona, koi hill nahi.

Topic ko yeh kyun chahiye. Global max/min claim karne ke liye tumhe critical points aur dono endpoints compare karne honge. Woh comparison hai Closed Interval Method (Global Extrema).

8. Do variables aur constraint — $g(x,y)=0$

Kabhi kabhi quantity do free numbers, $x$ aur $y$, par depend karti hai, likha jaata hai $f(x,y)$. Ek constraint ek extra rule hai jo unhe obey karna hota hai, likha jaata hai $g(x,y)=0$ — jaise "$x+y=10$" ko rearrange karo "$x+y-10=0$." Yahan $g$ ek doosra function hai jiska zero-set woh curve hai jis par tum chalne ke liye allowed ho.

Symbol $\nabla$ (ulta triangle, "nabla") prime mark ka do-variable cousin hai: yeh saare slopes ek saath collect karta hai.

Topic ko yeh kyun chahiye. Constraint ke saath tum freely move nahi kar sakte — tum curve $g=0$ par stuck ho. Tum $f$ ko tab tak improve kar sakte ho jab tak uska uphill arrow $\nabla f$ curve ke seedha cross point na kare, jahan uske saath kuch gain karne ko bacha nahi. Yahi Lagrange Multipliers (Multivariable) ka geometric seed hai aur AM-GM Inequality se connect karta hai.

9. Taylor expansion — polynomial se zoom in karna

Ek point $c$ ke paas likha: $f(c+h)\approx f(c)+f^{\prime }(c)h+\frac{1}{2}{f}^{\prime \prime }(c)h^2.$ Yahan $h$ hai "$c$ se ek tiny step door." Jab $f^{\prime }(c)=0$ ho, beech wala term disappear ho jaata hai aur change $\frac{1}{2}{f}^{\prime \prime }(c)h^2$ se govern hota hai — aur kyunki $h^2\ge 0$, $f^{\prime \prime }(c)$ ka sign sab decide karta hai. Yahan se classification test aata hai: dekho Taylor Series Expansion.

Prerequisite map

Ise aise padho: functions tumhe peaks aur valleys ka notion dete hain; slope derivative tak le jaata hai; derivative critical points deta hai aur (dobara differentiate karke) second derivative; endpoints, gradient aur Taylor sab final optimization machinery ko feed karte hain.

Equipment checklist

Right side cover karo aur parent note mein jaane se pehle khud ko test karo.

Connections

• Parent: Optimization
• Critical Points & Fermat's Theorem
• Second Derivative & Concavity
• Taylor Series Expansion
• Closed Interval Method (Global Extrema)
• Lagrange Multipliers (Multivariable)
• AM-GM Inequality
• Related Rates