1
vote
2answers
19 views

Trouble seeing how Lagrange Multipliers are True

So if a function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ constrained to the surface $g(x)=c$ for $x\in\mathbb{R}^n$ has a local maximum at $P$, then I'm having trouble seeing how this implies that the ...
2
votes
3answers
66 views

Lagrange multipliers from hell

I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way: "Find the closest and furthest points on the circle made from the intersection of the ...
0
votes
0answers
14 views

Maximizing distance between a region and a point with a Lagrange multiplier

I have a problem from an old midterm exam that I'm trying to complete so I can practice for the midterm tomorrow. Use the Lagrange multiplier method to find the points on the ellipse $x^2$ + ...
0
votes
1answer
24 views

Simple Lagrange Multiplier Problem, not working out

The question should be simple. Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for ...
0
votes
2answers
29 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
1
vote
1answer
32 views

Use Lagrange Multipliers to show the distance from a point to a plane

I'm trying to use Lagrange multipliers to show that the distance from the point (2,0,-1) to the plane $3x-2y+8z-1=0$ is $\frac{3}{\sqrt{77}}$. Our professor gave us two hints: We want to minimize a ...
1
vote
1answer
35 views

Lagrange multiplier - Find maximum on surface

I need someone to walk me through a 3 variable lagrange problem, since I haven't been able to find a reliable source to teach me, please. Here it is: Find the maximum of the function $F(x,y,z) = ...
0
votes
2answers
38 views

Extrema of two variable function

Find extrema of $f(x,y)=x^2-xy+y^2$ from set $M=\{ [x,y] \in \mathbb{R}^2;|x|+|y|\le1\}$ I am solving this kind of problems for the first time and I am not sure what I am doing, what I have got ...
-1
votes
1answer
41 views

Optimization: Finding line connecting non-pareto-optimal allocation in Edgeworth Box to PO allocation

Two people, A and B, with respective utility functions of: $$U_a(X_a,Y_a) = X_a^2 Y_a\\ U_b(X_b,Y_b) = X_b Y_b^2$$ Total $X$ (that is, $X_a+X_b$) is fixed at $36$. Total $Y$ ($Y_a+Y_b$) is fixed ...
-1
votes
3answers
61 views

Lagrange Multiplier with $3$ variables

Here is the question: $f(x,y,z) = x + 2y^2 - 3z$ subject to the constraint $z = 4x^2 + y^2$. I don't understand how to do this because when I take the partial derivative in respect to $z$, I get $-3 ...
0
votes
0answers
30 views

Lagrange multiplier uses

I'm currently self studying calculus of variations and stumbled upon something called a Lagrange multiplier, used in solving isoperimetric problems and the like. Not knowing what they are I backtrack ...
2
votes
3answers
76 views

Using LaGrange multipliers to solve for minimum

I am having troubles with one part of this homework problem. Hopefully somebody can help me out: Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the given constraint ...
0
votes
1answer
64 views

How do I maximize the function $W=\min(u_1,u_2)$ subject to the constraint $u_1^2+u_2^2=100$?

How do I maximize the function $W=\min(u_1,u_2)$ subject to the constraint $u_1^2+u_2^2=100$? Usually I would do a Langrangian on the function to be maximized, but here it is difficult to do so. The ...
1
vote
0answers
14 views

How do I find the maximum and minimum values of xy on an off-center ellipse?

What's the maximum and minimum of f(x,y)=xy with the constraint $$\dfrac{(x-x_o)^2 }{A^2} + \dfrac{(y-y_o)^2 }{B^2} = 1$$ Using lagrangian multipliers is simple when the center of the ellipse is ...
0
votes
1answer
31 views

Lagrangian Multiplier Question

I can do question 2 easily but I'm running into some problems proving 1 rigorously. No idea how to go about doing it at all.
2
votes
1answer
69 views

Using Lagrange multipliers to maximize function

Use Lagrange multipliers to maximize function $$f(x,y)=6xy,$$ subject to the constraint $$2x+3y=24.$$ $$F(x,y,\lambda)=6xy+\lambda(2x+3y-24)$$ $$F_{x}=6y+2\lambda=0$$ $$F_{y}=6x+3\lambda=0$$ ...
1
vote
1answer
38 views

How can I calculate the maximum and minimum of this function?

This is incorrect and I have a feeling I am not doing this correctly.
1
vote
1answer
36 views

Let $f(x,y)=\frac {x^2}{2}+\frac {y^2}{4}$ on $\{(x,y)|x^2-y^2=2\}$. Find the absolute maximum and minimum if they exist.

Let $f(x,y)=\frac {x^2}{2}+\frac {y^2}{4}$ on $\{(x,y)|x^2-y^2=2\}$. Find the absolute maximum and minimum if they exist. I approached this problem using lagrange multiplier, with $g(x,y)=x^2-y^2-2$. ...
2
votes
0answers
63 views

Find maximum of a double integral over a region

I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2*(3ab^3+12a^3b-6a^3b^2)*\sin^2({\pi ax + \pi by}})dxdy$$ I need to find the values of $a$ and $b$ that ...
1
vote
1answer
79 views

Minimization problems (using Lagrange Multipliers/Directional Derivatives)

The first problem asks to find the minimum cost of a rectangular box if the bottom costs \$2, the sides \$5 per square foot, and the top costs $7 per square foot. The volume of the box is given as 20 ...
1
vote
3answers
37 views

Constrained optimisation question

Since $f$ has a local extremum at $x_1$, then surely the LHS of equation (3) always zero? If so, then isn't lambda always simply zero too? But this cannot be, otherwise the last sentence of the ...
0
votes
0answers
39 views

How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
1
vote
1answer
41 views

How do I find the highest and lowest points made by the union of these two functions using Lagrange Multipliers?

Find the highest and lowest points made by the union of these two functions using Lagrange Multipliers. $x^2+y^2+z^2 = 16$ $(x+1)^2+(y+1)^2+(z+1)^2 = 27$ I got the basics down, I used the first ...
1
vote
1answer
46 views

Deriving the equations in a lagrange multiplier

I was going though this website for a general idea of Lagrange Multipliers. One of its examples was to maximize $xyz$ given $xy+yz+zx= 32$. The first few lines of the solution went as follows: $$ $$ ...
0
votes
2answers
50 views

Problem on Lagrange multipliers

This problem has two parts: $a)$ Let $k>0$, find the minimum of the function $f(x,y)=x+y$ over the set S=$\{(x,y) \in \mathbb R^2_{> 0}:xy=k\}$. $b)$ Prove that for every $(x,y) \in \mathbb ...
1
vote
2answers
100 views

Applying the Lagrangian function to find critical points

So I have the following function $$ f(x,y) = x^2+y^2 $$ subject to $$ g(x,y) = x+y-1 = 0. $$ And I have to use the Lagrangian to find the critical points, and determine wether they are ...
1
vote
1answer
112 views

Determine all the extrema of a function subject to a non-linear constraint.

QUESTION Determine all extrema of the function $$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$ ATTEMPT I don't think I understand what I'm supposed to do. This was in a test and I ended up ...
0
votes
1answer
75 views

Finding the shortest distance between a point and a coplex surface

I have a surface which is $z=ax+bx^2+cxy+d$, where $a,b,c$ are coefficients and $d$ is the constant. so an arbitrary point would be $(x, y, ax + bx^2 + cxy+ d)$. there are a set of points that are not ...
0
votes
1answer
239 views

Find all extrema for $f(x,y) = 3xy$ subject to the constraint $4x^2 + 2y = 48$

Find all extrema for $f(x,y) = 3xy$ subject to the constraint $4x^2 + 2y =48$. I put it into the form of: $3xy - \lambda(4x^2 +2y -48) = F(x,y,\lambda)$ $3xy - 4x^2\lambda -2y\lambda + 48\lambda$ ...
1
vote
2answers
842 views

Simple explanation of lagrange multipliers with multiple constraints

I'm studying support vector machines and in the process I've bumped into lagrange multipliers with multiple constraints and Karush–Kuhn–Tucker conditions. I've been trying to study the subject, but ...
2
votes
3answers
167 views

Lagrange multiplier problem of looking for the point on $\frac1x + \frac1y + \frac1z =1$ closest to the origin

Use Lagrange multipliers to find the point on the surface $$\frac1x + \frac1y + \frac1z =1$$ which is closest to the origin. I was wondering if I would start off by using the distance formula, ...
2
votes
1answer
85 views

What justifies assuming that a level surface contains a differentiable curve?

My textbook's proof that the Lagrange multiplier method is valid begins: Let $X(t)$ be a differentiable curve on the surface $S$ passing through $P$ Where $S$ is the level surface defining the ...
1
vote
2answers
61 views

a problem using Lagrange multipliers

Prove that $ \frac{{n!}} {{n^{\frac{n} {2}} }} $ is the max of the function $ f\left( x \right) = \prod\limits_{i = 1}^n {x_i } $ under the restriction $ g\left( x \right) = \sum\limits_{i = 1}^n ...
0
votes
1answer
174 views

a basic problem about lagrange multipliers

Find the max and min values of the function $f(x,y)=x^2+y^2$ under the restriction $g(x,y)=\frac{x^2}{2}+y^2-1=0 $ Note that we can use Lagrange Multipliers Theorem , since $grad(g(p))\ne 0$ $\forall ...
6
votes
3answers
230 views

Lagrange multiplier method, find maximum of $e^{-x}\cdot (x^2-3)\cdot (y^2-3)$ on a circle

I attempted to design an exercise for my engineer students and couldn't solve it myself. Maybe here are some experts in calculus who have some better tricks than I do: The exercise would be to ...
2
votes
1answer
117 views

Lagrange multiplier constrain critical point

When using Lagrange multipliers in an inequelity, $$ f(x,y) = x^2+y $$ with the constraint $$ x^2+y^2 \leq 1. $$ I have to find the critical points inside the "disk" right? I've done $$ f_x = 2x ...
2
votes
1answer
68 views

Solving Lagrange multipliers system

I need help solving this system: $$ \begin{cases} 2(x-1) = \lambda2x \\ 2(y-2) = \lambda2y \\ 2(z-2) = \lambda2z \\x^2 + y^2+z^2 = 1 \end{cases} $$ I can find $$ \lambda = (x-1)/x $$ but can't go ...
3
votes
2answers
129 views

How to use Lagrange Multipliers, when the constraint surface has a boundary?

The method called Lagrange Multipliers is used to find critical points of $f(x_1,x_2,\ldots,x_n)$, when $f$ is constrained to the level set $S = \{ x\in \mathbb{R}^n \, | \, g(x_1,x_2,\ldots,x_n)=0 ...
1
vote
2answers
95 views

Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.

This is not homework, but it is in my book and I find it hard to solve: Determine the points where $f$ is has a local minimum/maximum. Determine if it strong/weak and absolute/relative and ...
0
votes
2answers
215 views

Solution to a system of symmetric equations

After applying the Lagrange multiplier method, I got the following system of equations, which is quite symmetric: $(x+y)^2 + (x+z)^2 = \frac{2}{3} \lambda x$ $(y+x)^2 + (y+z)^2 = \frac{2}{3} \lambda ...
0
votes
1answer
83 views

Lagrange multipliers for x,y,z

I have this question, I have run completely blind into. Find by Lagrange multipliers the volume V=xyz of where the largest box with sides adding up to x+y+z = k. I have found the gradient of V: ...
1
vote
1answer
307 views

What is the intuition behind the Lagrange multiplier?

I know that the minimum or maximum point is achieved when the gradient in the constraint function is parallel to the gradient on the $f$ function. But why the Lambda is called the Lagrange ...
1
vote
4answers
835 views

Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant. *help needed please*

Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant. This is what i've done so far.... Let $a,b,c$ be some points that the ...
2
votes
2answers
7k views

Find Max/Min volume of rectangular box using Lagrange Multipliers

Can anyone help me solve the problem below? This is question number 14.8.42 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the maximum and minimum volumes of a ...
1
vote
1answer
3k views

Use LaGrange multipliers to find maximum and minimum values

I am having trouble understanding how to solve the problem below. Can anyone show me how to solve this? Here is the problem definition: "Use LaGrange multipliers to find the maximum and minimum ...