Lagrange multipliers Definition and 173 Threads

Homework Statement A ray of light enters a glass block of refractive index n and thickness d with angle of incidence θ1. Part of the ray refracts at some angle θ2 such that Snell's law is obeyed, and the rest undergoes specular reflection. The refracted ray reflects off the bottom of the block...

Homework Statement Lagrange multipliers to find the maximum and minimum values of f(x,y) = 4x^3 + y^2 subject to the constraint 2x^2 + y^2 = 1. Find points of these extremum. Homework Equations The Attempt at a Solution g(x,y)= 2x^2 + y^2 - 1 f(x,y)= 4x^3 + y^2 Gradient F=...

Here is the question: I have posted a link there to this topic so the OP can view my work.

I'm stuck on this problem for the past hour. I've tried solving for all variables and none of the steps I'm doing are getting me to the right answer. Find the point on sphere x^2+y^2+z^2=25 farthest from point (1,1,-1). My steps: http://i.imgur.com/c5kUj9g.png Correct Answer: x=...

Homework Statement Show that the largest rectangular box having a fixed surface area must be a cube. Homework Equations ##V(x,y,z) = xyz## ##\sigma(x,y,z) = 2(xy + yz + zx) = C \in \mathbf{R}## The Attempt at a Solution As of this assignment, we haven't yet learned Lagrange...

Find the minimum value of \(\int_0^1y^{'2}dx\) subject to the conditions \(y(0) = y(1) = 0\) and \(\int_0^1y^2dx = 1\). Let \(f = y^{'2}\) and \(h = y^2\). Then \begin{align*} G[y(x)] &= \int_0^1[f - \lambda h]dx\\ &= \int_0^1\left[y^{'2} - \lambda y^2\right]dx \end{align*}...

f(x,y)=x^2y with the constraint of x^2+2y^2=6 Use lagrange multipliers to find the extrema. Thanks!

Here is the question: Here is a link to the question: Please help with lagrange multipliers? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Hi we covered the Lagrange multiplier method in Lagrangian Mechanics and as far as I know, is the physical meaning behind this to be able to solve either some non-holonomic constraints or to get some information about the constraint forces. my problem is, i do not know the physical meaning of...

Homework Statement Find max/min of f subject to constraint: x^2+y^2+z^1 = 1 Homework Equations f(x,y,z) = 1/4*x^2 + 1/9*y^2 + z^2 g(x,y,z) = x^2 + y^2 + z^2 - 1 The Attempt at a Solution L = 1/4*x^2 + 1/9*y^2 + z^2 - λ(x^2 + y^2 + z^2 - 1) Lx = 2/4*x - λ*x*2 Ly = 2/9*y -...

Here is the question: Here is a link to the question: Calc 3 Lagrange multiplier question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Problem stated: Let \(a_1, a_2, ... , a_n\) be \(n\) positive numbers. Find the maximum of $$\sum_{i=1}^{n}a_ix_i$$ subject to the constraint $$\sum_{i=1}^{n}x_i^2=1$$. I honestly have not much of an idea of how to go about solving this. If I use lagrange multipliers which I think I am supposed...

The problem given is to find the local extreme values of \(f(x,y)=x^2y\) on the line \(x+y=3\). I went through the system of equations with the partial derivatives of \(x\), \(y\), and \(\lambda\), and found two extreme points \((0,3)\) and \((2,1)\). Plugging that into the original function I...

Homework Statement Here is the problem, the solution and my question (in red): I'm guessing it was rejected because for the volume function, the dimensions cannot be negative? What if it was not volume and instead was just an arbitrary function. In that case you would not reject...

Advanced Calculus of Several Variables, Edwards, problem II.4.1: Find the shortest distance from the point (1, 0) to a point of the parabola y^{2} = 4x. This is the Lagrange multipliers chapter. There might be another way to solve this, but the only way I'm interested in right now is the...

Given the equations $$ rv\cos\gamma - h = 0,\quad \frac{v^2}{2} - \frac{\mu}{r} + \frac{\mu}{2a} = 0 $$ I want to maximize gamma. Do I have to solve for gamma in the first equation to use the method of Lagrange multipliers, or if not, how would I do this in the current form?

Homework Statement Use Lagrange multipliers to find the eigenvalues and eigenvectors of the matrix A=\begin{bmatrix}2 & 4\\4 & 8\end{bmatrix} Homework Equations ... The Attempt at a Solution The book deals with this as an exercise. From what I understand, it says to consider...

I was wondering how they got all the different combinations of points? Why can't they just put (+-√2,+-1,+-√(2/3)) ?

Homework Statement Find the maximum and minimum values of 2x2 + y2 on the curve x2 + y2 - 4x = 5 by the method of Lagrange Multipliers. Homework Equations I will express my Lagrange multipliers as λ. The Attempt at a Solution Okay so we want the max min of f(x,y) = 2x2 + y2 given...

Homework Statement If n is a fixed positive integer, compute the max and min values of the function (x-y)^n = f(x,y), under the constraint x^2 + 3y^2 = 1 The Attempt at a Solution I got the 4 critical points (±\frac{\sqrt{3}}{2}, ±\frac{1}{2\sqrt{3}})\,\,\text{and}\,\...

I understand that for Lagrange multipliers, ∇f = λ∇g And that you can use this to solve for extreme values. I have a set of questions because I don't understand these on a basic level. 1. How do you determine whether it is a max, min, or saddle point, especially when you only get one...

Homework Statement Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f(x,y) = exy; g(x,y) = x3 + y3 = 16 Homework Equations ∇f(x,y) = λ∇g(x,y) fx = λgx fy = λgy The Attempt at a Solution ∇f(x,y) = < yexy, xexy > ∇g(x,y) = <...

In Goldstein, the action is defined by I=\int L dt. However, when dealing with constraints that haven't been implicitly accounted for by the generalized coordinates, the action integral is redefined to I = \int \left( L + \sum\limits_{\alpha=1}^m \lambda_{\alpha} f_a \right) dt. f is...

Normally lagrange multipliers are used in the following sense. Suppose we are given a function f(x,y.z..,) and the constraint g(x,y,z,...,) = c Define a lagrange function: L = f - λ(g-c) And find the partial derivatives with respect to all variables and λ. This gives you the extrema...

I have been reading a little about calculus of variations. I understand the basic method and it's proof. I also understand Lagrange multipliers with regular functions, ie since you are moving orthogonal to one gradient due to the constraint, unless you are also moving orthogonal to the other...

When I try to solve a linear program using matlab,after using linprog(f,A,b,...) I can find the Lagrange multiplier associated with the inequality constraints and the lower bound constraints by using: lambda.ineqlin ; lambda.lower But if I want to solve a quadratic program (using...

Homework Statement I need to find the extrema of f(x,y) = 3x^{2} + y^{2} given the constraint x^{2} + y^{2} = 1 Homework Equations I'm not sure what goes here. I've been trying to solve it with this: ∇f(x,y) = λ∇g(x,y) The Attempt at a Solution f(x,y) = 3x^{2} + y^{2} g(x,y)...

Homework Statement Show that , the maximum value of function f(x,y) = x^2 + y^2 is 70 and minimum value is 20 in constraint below. Homework Equations Constraint : 3x^2 + 4xy + 6y^2 = 140 The Attempt at a Solution Book's solution simply states the Lagrange rule as ...

Homework Statement What I don't understand is why you can maximize the distances squared - d2. Isn't d2 different from d? I don't see how they can get you the same value.

Homework Statement The problem of minimizing f(x1, x2) = x1^3 subject to (x1 + 1)^3 = (x2 − 2)^2 is known to have a unique global solution. Use the method of Lagrange multipliers to find it. You should deal with the issue of whether a constraint qualification holds. Homework Equations...

Does anyone have any tips for solving the system of equations formed while trying to find Lagrange Multipliers? I have searched for videos online (patrickjmt and the MIT lecture on Lagrange Multipliers) but I still find it a bit confusing.

Homework Statement An open gutter with cross section in the form of a trapezoid with equal base angles is to be made by bending up equal strips along both sides of a long piece of metal 12 inches wide. Find the base angles and the length of the sides for maximum carrying capacity. For more...

Homework Statement Attached as Question.jpg. Homework Equations Partial differentiation. Lagrange multiplier equation. The Attempt at a Solution Attached as MyWork.jpg. Is my work correct? I'm still not confident with myself for these problems and it would be great if someone...

Homework Statement When a rectangular box is sent through the mail, the post office demands that the length of the box plus twice the sum of its height and width be no more than 250 centimeters. Find the dimensions of the box satisfying this requirement that encloses the largest possible...

Homework Statement Not really a homework question, just want to check out if what I'm doing is right. I challenged myself to find the equation of motion and the forces in the simple pendulum system but with using the Lagrange multipliers and the constraint equation.Homework Equations In next...

Homework Statement Hello! I'm having some difficulty getting the objective function out of this question, any help/hints would be appreciated >.< Company A prepares to launch a new brand of tablet computers. Their strategy is to release the first batch with the initial price of p_1 dollars...

The problem: Minimize tr{RyxR} subject to RTR=I This problem is known as Procruses Analysis and can be solved using Lagrange Multipliers, so there's a tendency to write the following function: L(R) = tr{RyxR} - \Lambda(RTR-I), where \Lambda is a matrix of Lagrange Multipliers However, there...

Homework Statement A cannonball is heated with with temperature distribution T(x,y,z)=60(y2+z2-x2). The cannonball is a sphere of 1 ft with it's center at the origin a) Where are the max and min temperatures in the cannonball, and where do they occur?Homework Equations \nablaf=λ\nablag Where...

In Lagrangian mechanics, can anyone show how to find the extrema of he action functional if you have more constraints than degrees of freedom (for example if the constraints are nonholonomic) using Lagrange Multipliers? I've looked everywhere for this (books, papers, websites etc.) but none...

Hello everyone, i have 2 problems in my multivariable calculus homework that i can't solve . Please help me out, thank you so much 1/f(x,y)= [(x^2) -2y]^(0.5) a) Find directional derivatives of f at (2,-6) in the direction of <-4,3> b) Find equation of the tangent plane to the function...

Homework Statement Use Lagrange multipliers to find the max and min values of the function subject to the given constraints: f(x,y,z)= x2y2z2 constraint: x2 + y2 + z2 = 1 Homework Equations ∇f = ∇g * λ fx = gx * λ fy = gy * λ fz = gz * λ The Attempt at a Solution i can't solve...

Homework Statement Use Lagange Multipliers to find the max and min values of the function subject to the given constraint(s). f(x,y)=exp(xy) ; constraint: x^3 + y^3 = 16 Homework Equations \nablaf = \nablag * \lambda fx = gx * \lambda fy = gy * \lambda The Attempt at a Solution...

Homework Statement I am trying to find the min and max values of f(x,y)=2x^2 + 3y^2 subject to xy=5. Homework Equations f(x,y)=2x^2 + 3y^2 subject to xy=5 \mathbf\nablaf=(4x, 6y) \mathbf\nablag=(y,x) The Attempt at a Solution When I go through the calculations, I end up with two critical...

Homework Statement Use Lagrange multipliers to ¯nd the maximum and mini- mum value(s), if they exist, of f(x; y; z) = x^2 -2y + 2z^2 subject to the constraint x^2+y^2+z^2 Homework Equations The Attempt at a Solution Basically after I find the gradient of the functions I get this. 2x=2x lamda...

My textbook is using Lagrange multipliers in a way I'm not familiar with. F(w,λ)=wCwT-λ(wuT-1) Why is the first order necessary condition?: 2wC-λu=0 Is it because: \nablaF=2wC-λu Why does \nablaF equal this? Many thanks! Edit: C is a covariance matrix

Dear all, I have an optimization problem with boundary conditions, the type that is usually solved with Lagrange multipliers. But the (many) variables my function depends on can take only the values 0 and 1. Does anyone know how to apply Lagrange multipliers in this case? I am a...

So I need to find the min and max values of f(x,y,z) = x^2 + 2y^2 + 3z^2 given the constraints x + y + z = 1 and x - y + 2z =2. I've gotten as far as (2x, 4y, 6z) = (u,u,u) + (m,-m,2m). I'm stuck trying to solve this system of equations. Any hints?

URGENT - Lagrange Multipliers Homework Statement :confused: Using the method of lagrange multipliers prove the formula for the distance from a point (a,b,c) to a plane Ax + By + Cz = DThe Attempt at a Solution Using the equation of the form; H(x,y,z,L) = (x-a)^2 + (y -b)^2 +(z-c)^2 + L(Ax...

1. Use the method of Lagrange multipliers to nd the minimum value of the function: f(x,y,z) = xy + 2xz + 2yz subject to the constraint xyz = 32. I understand the method how Lagranges Multipliers is donw done but seem to have got stuck solving the Simultaneous Equations involving the...

Homework Statement Find the point closest to the origin on the line of intersection of the planes y + 2z = 12 and x + y = 6Homework Equations \nuf = \lambda\nug1 +\mu\nug2 f = x2+y2+z2 g1: y + 2z = 12 g2: x + y = 6 There are supposed to be gradients on all of those, whether or not LaTeX...