14.8 Lagrange Multipliers PDF

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14.8 Lagrange Multipliers Saturday, February 13, 2021

10꞉44 AM

Lagrange Multipliers ∇f(v%&) = λ∇g(v%&) -

Method for maximizing 𝑓(𝑥, 𝑦, 𝑧) subject to 𝑔 (𝑥, 𝑦, 𝑧 ) = 𝑘 . Find maximal f(v)=c within the surface g(v)=k; ○ This occurs at v where f(v)=c is tangent to g(v)=k § Tangent: gradient vectors are || ○ where ∇𝑓 (𝑣- ) = 𝜆∇𝑔(𝑣-) where3lambda3is3a3scalar

The Method 1.

2.

Find 𝑥, 𝑦, 𝑧, 𝜆 such3that a. ∇𝑓(𝑥, 𝑦, 𝑧 ) = 𝜆∇𝑔 (𝑥, 𝑦, 𝑧 ) i. 𝑓! (𝑥, 𝑦, 𝑧) = λg" (x, y, z) etc. b. 𝑔(𝑥, 𝑦, 𝑧) = 𝑘 Evaluate f at all returned points to find max/min.

Example Maximize the volume of a lidless box made from 12m^2 of cardboard.

**Create the function and its constraint. 𝑉(𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧 𝑆(𝑥, 𝑦, 𝑧) = 2𝑥𝑧 + 2𝑦𝑧 + 𝑥𝑦 = 12

**Find x, y, z, lambda for the lagrange multiplier equation. ∇𝑉(𝑥, 𝑦, 𝑧) = 𝜆∇𝑆 (𝑥, 𝑦, 𝑧 ) 𝑉! = 𝜆𝑆! 𝑦𝑧 = 𝜆(2𝑧 + 𝑦) 𝑉# = 𝜆𝑆# 𝑥𝑧 = 𝜆(2𝑧 + 𝑥) 𝑉$ = 𝜆𝑆$ 𝑥𝑦 = 𝜆 (2𝑥 + 2𝑦 ) 𝑆 = 12 2𝑥𝑧 + 2𝑦𝑧 + 𝑥𝑦 = 12

**We can then solve the system using our intuition

We can then solve the system using our intuition… [3𝑦𝑧 = 𝜆(2𝑧 + 𝑦)3 ] ∗ 𝑥 etc.%will%make%them%all%=%xyz 𝑥𝑦𝑧 = 𝑥𝜆(2𝑧 + 𝑦 ) = 𝑦𝜆 (2𝑧 + 𝑥 ) = 𝑧𝜆 (2𝑥 + 2𝑦 ) 𝑥𝑦𝑧 = 𝜆(2𝑥𝑧 + 𝑥𝑦 ) = 𝜆(2𝑦𝑧 + 𝑥𝑦 ) = 𝜆 (2𝑥𝑧 + 2𝑦𝑧 ) 2𝑥𝑧 + 𝑥𝑦 = 2𝑦𝑧 + 𝑥𝑦 2𝑥𝑧 = 2𝑦𝑧 𝑥=𝑦 2𝑦𝑧 + 𝑥𝑦 = 2𝑥𝑧 + 2𝑦𝑧 𝑥𝑦 = 2𝑥𝑧 𝑦 = 2𝑧 2𝑥𝑧 + 𝑥𝑦 = 2𝑥𝑧 + 2𝑦𝑧 𝑥𝑦 = 2𝑦𝑧 𝑥 = 2𝑧 → 𝑥 = 𝑦 = 2𝑧 2𝑥𝑧 + 2𝑦𝑧 + 𝑥𝑦 = 12 = 4𝑧 % + 4𝑧 % + 4𝑧 % 4 = 𝑧% 𝑧=1 → 𝑥 = 2, 𝑦 = 2

**Done.

WebAssign 1.V

𝑓(𝑥, 𝑦 ) = 60, 30

2.V

∇𝑓(𝑥, 𝑦 ) = 𝜆∇𝑔(𝑥, 𝑦 ) 𝑓! = 𝜆𝑔! = 6 = 𝜆2𝑥 3 =𝑥 𝜆 𝑓# = 𝜆𝑔# = 6 = 𝜆2𝑦 3 =𝑦 𝜆 →𝑥=𝑦 % 𝑥 + 𝑦 % = 18 → (3,3 ), (−3, −3 ) 𝑓(3,3 ) = 18 + 18 = 36 𝑓(−3, −3 ) = −18 − 18 = −36...