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4.4.15 · D5 · HinglishMultivariable Calculus

Question bank — Lagrange multipliers — one and two constraints

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4.4.15 · D5 · Maths › Multivariable Calculus › Lagrange multipliers — one and two constraints

Shuru karne se pehle ek vocabulary reminder (har symbol yahan earn kiya gaya hai taaki koi undefined symbol kabhi na mile):

  • — woh cheez jo hum optimize karte hain (ek tarah ki "height"). Iska gradient woh arrow hai jo batata hai ki kis direction mein sabse fast badhta hai. Dekho Gradient and directional derivative.
  • , — constraints: woh surfaces (ya unke cross karne se bani curve) jin par rehna zaroori hai. , woh arrows hain jo un surfaces se seedhe bahar nikalte hain — yaani normals. Dekho Tangent planes and normal vectors.
  • (lambda), (mu) — multipliers: woh numbers jo batate hain ki ko reproduce karne ke liye kitna-kitna normal arrow mix karna hai.
  • — ek tangent direction: woh direction jisme tum actually chal sakte ho aur feasible rehte ho.

Neeche sab kuch parent method ko har angle se probe karta hai.

True or false — justify

The unit circle example: max of is
True. par, ; sign-flipped multiplier deta hai aur minimum milta hai, toh dono cases cover ho jaate hain.
At a constrained maximum, must be the zero vector
False. Constraint par usually nonzero hota hai — bas use surface se seedha bahar point karna hota hai ( ke parallel), taaki koi allowed direction ko badha na sake.
akele optimal point locate kar leta hai
False. Yeh equation sirf direction match fix karti hai; tumhe abhi bhi chahiye taaki pata chale ki surface par kahan ho — isliye equations hote hain.
If at a solution, the constraint is doing nothing there
True. se milta hai — yeh ek ordinary unconstrained critical point hai jo ittifak se surface par baith gaya; constraint ko relax karne se nahi badlega.
A large positive means the constraint is "cheap" to relax
False. Large ka matlab hai ki relaxation ke ek unit se mein bada improvement milta hai, toh constraint expensive / hard binding hai — yahi shadow price reading hai.
For two constraints in , must equal either or
False. Yeh ek combination hona chahiye: us poore plane mein leta hai jo dono normals span karte hain, sirf ek mein nahi.
Lagrange's method automatically finds global maxima
False. Yeh sirf candidate stationary points dhundta hai; global max ya min decide karne ke liye unke -values compare karne hote hain (aur singular points / boundaries check karne hote hain).
is always perpendicular to the surface
True (jahan ). differentiate karne par milta hai har tangent ke liye, toh sabhi tangents ko zero kar deta hai — yahi normal hai. Dekho Level sets and contours.
The multiplier value can legitimately be negative
True. Iska sign batata hai ki constraint relax karne par kis direction mein jaata hai; min problem ya alag-oriented sign flip kar deta hai, jaise milta hai ke liye circle par.

Spot the error

"To minimize on , I set and solved."
Galat tool: se unconstrained critical points milte hain aur fence ignore ho jaati hai. Tumhe plus constraint chahiye.
"I wrote and got ; done."
Incomplete: tumne kabhi use hi nahi ki, toh point located nahi hua. Constraint equation drop karna classic under-determination error hai.
"On the intersection of two surfaces I used only ."
Tumne doosri surface bhool gayi. Feasible set ek curve hai, toh ko dono normals se balance karna hoga: .
" and point the same way, so always."
Aisa nahi. Parallel mein antiparallel bhi included hai; mein allowed hai jab dono arrows opposite direction mein point karein.
"The tangency of level curves means and have equal values there."
Nahi. Tangency ka matlab hai unki level curves touch karti hain (parallel normals), na ki . Yeh directions ka geometric alignment hai, function values ka nahi. Dekho Level sets and contours.
"Since Lagrange gave one point, that's the answer."
Method kayi stationary points de sakta hai (min, max, saddle on the curve); algebra ka ek root poori kahani kabhi nahi hoti. Hamesha enumerate karo aur compare karo.
"At a singular point where , the equation still pins things down."
Yeh ki parwah kiye bina mein collapse ho jaata hai, toh method koi information nahi deta; aisi points ko alag se check karna hoga — Lagrange unhe silently miss kar sakta hai.
"Two constraints in give 3 equations."
Miscount. 3 gradient equations + 2 constraint equations = 5 equations in 5 unknowns hote hain.

Why questions

Why must be normal to the constraint surface at an extremum?
Agar usmein koi tangential component hota, toh tum surface ke along us direction mein chal ke badal sakte the — toh woh extremum nahi hota. Sirf ek purely normal hi improve nahi kiya ja sakta.
Why does live in the plane spanned by for two constraints?
Ek allowed tangent dono normals ke perpendicular hai; force karta hai ki mein koi -part na ho, yaani woh us plane mein ho jo dono normals define karte hain.
Why is called a "shadow price"?
Kyunki jahan constraint level hai: yeh constraint ko loosen karne per unit optimum mein marginal improvement hai — literally iska economic price. Dekho Dual problem and shadow prices.
Why do we differentiate along curves inside the surface?
Kyunki constraint par hum sirf aise curves ke along move kar sakte hain; constrained extremum har feasible path par ka ordinary extremum hai, jo deta hai.
Why does the chain rule turn into ?
Chain rule kehta hai ki ki rate of change ke along, gradient aur velocity ka dot product hai; use zero set karna exactly extremum condition hai. Dekho Gradient and directional derivative.
Why doesn't the sign or magnitude of affect where the optimum is?
sirf dono gradient arrows ki lengths balance karta hai; location parallel directions require karne aur satisfy karne se fix hoti hai.
Why does Lagrange generalize to KKT conditions for inequalities?
Ek inequality ya toh slack hoti hai (multiplier , constraint idle) ya tight (equality ki tarah behave karti hai sign-restricted multiplier ke saath); KKT dono cases ko usi gradient-balance idea mein bundle karta hai.

Edge cases

What if at a candidate point?
Surface ka wahan koi well-defined normal nahi hota (singular point); ban jaata hai kisi bhi ke liye, toh us point ko haath se inspect karna hoga — Lagrange wahan extrema miss kar sakta hai.
What if and are parallel (dependent) at a point?
Woh plane span nahi karte, toh intersection clean curve nahi rahi; two-constraint setup degenerate ho jaata hai aur uniquely determined nahi hote — constraint-qualification failure.
What if the constraint set is a closed region with a boundary, not just ?
Lagrange sirf smooth surface search karta hai; extrema kisi boundary edge par baith sakte hain jo describe nahi karta, toh un edges ko alag check karna hoga.
What happens to the method as the constraint is removed (no )?
Tum wapas par aa jaate ho — unconstrained case, jo exactly limit hai jahan constraint koi pull nahi exert karta.
Can a constrained problem have no Lagrange solution yet still have a max?
Haan — agar max singular point par ya unincluded boundary par ho, toh gradient-balance equations ka wahan koi root nahi hota, toh woh miss ho jaata hai jab tak alag examine na karo.
Why does the paraboloid-plus-plane problem give two roots for ?
Intersection curve neeche jaati hai phir uthti hai, toh ka us par ek minimum aur ek maximum dono hain — do stationary points, jo ke do roots se match karte hain.
What if but the point still lies on ?
Tab ke saath hold karta hai: yeh ek unconstrained critical point hai jo ittifak se feasible hai — perfectly valid (constraint-inactive) Lagrange solution.
What if is constant on the whole constraint surface?
Har feasible point trivially ek extremum hai; ya toh zero hai ya har jagah already normal hai, toh unconstrained/undetermined hai — problem degenerate hai.
Recall Fast self-check

ka ek feasible point par tangential component hai — extremum hai ya nahi? ::: Extremum nahi: tum abhi bhi us tangential component ke along step lekar badha sakte ho. Do parallel gradients hain par — valid Lagrange point? ::: Nahi: parallel gradients zaroori hain par tumhara surface par hona bhi zaroori hai, .

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