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0answers
22 views

Lagrange multipliers (distance)

Find the closest point of the surface $z=xy-1$ to the origin. How would you do that with Lagrange multipliers?
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0answers
26 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
1
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1answer
21 views

Unable to solve system of equations in Lagrange multiplier problem.

The problem: Find the right triangular prism of given volume and least area if the base is required to be a right triangle. As for parameters of the right triangular prism, $V$ is volume, $A$ is ...
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0answers
12 views

Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
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1answer
37 views

extrema of funcion

$f(x,y,z)=x+2z$ and $M=\{[x,y,z]\in\mathbb{R}^3:x^2+2y^2=4,z+y\le 1\}$. I found out that M is not bounded from below so it does not have minimum or infimum. But how do I find maximum? I tried to use ...
0
votes
1answer
58 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
1
vote
1answer
43 views

Lagrange's Multiplier Method

I need to find the distance between the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ and the line $y = 10 - 2x$ using Lagranges' Multiplier Method. So far I know how to find the minimum distance ...
4
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0answers
39 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
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1answer
71 views

The shortest path connecting three points

I have 3 points X,Y,Z, lets call them buildings. I need to find the shortest amount of path that connects the 3 buildings, these buildings can be in any sort of shape and any distance from each ...
1
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2answers
29 views

Does Lagrange multiplier have solution if functions doesn't intersect

I am trying to get intuition behind Lagrange multiplier and question that bothers me is: Does Lagrange multiplier have solution if two functions(main function and constraint) doesn't intersect. Thank ...
2
votes
1answer
30 views

Lagrange multipliers…what is my constraint?

How would I use Lagrange multipliers to determine which point on the surface $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ with $x,y,z>0$ is closest to the origin? I'm not sure what the constraint would ...
4
votes
2answers
66 views

Lagrange multiplier - space probe

i am stuck on this question which uses the Lagrange multiplier. I am trying to construct the equations using the partial derivatives but the $x$'s and $y$'s cancel. can anyone help? A space probe in ...
2
votes
3answers
98 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 ...
2
votes
3answers
92 views

Lagrange Method Problem

I am from engineering background and I am currently studying calculus. I had a question from assignment to be solved from a course on coursera but I could not do it. People have posted solution in the ...
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3answers
60 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 ...
0
votes
2answers
65 views

why is the derivative equal to zero here?

In this MIT open course video Denis Auroux said that given a function $f(x, y)$ and a restriction level surface $g(x, y) = c$, for any vector $u$ tangent to this surface $g=c$, we must have: $$ ...
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1answer
26 views

Can I use Lagrange Multipliers with inequality constraints?

Suppose I had a problem: Maximize $f(\bf{x}) $ subject to the contraints $g_i(\bf{x})< b$ Can I still use Lagrange multipliers? My text says that the constraints need to be equalities.
2
votes
2answers
173 views

Closest distance between two quadratic curves

I'm having trouble with the following problem : "find the closest distance between $x^2+4y^2=4$ and $xy=4$" I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted ...
0
votes
2answers
40 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 ...
0
votes
0answers
38 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 ...
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1answer
68 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 ...
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votes
1answer
34 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.
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0answers
29 views

How can I find the Min and max of this question?

I have been trying for the past 2 hours on this question and cannot seem to figure out the answer. So far I have gotten the 'green' bits correct. Someone Help please
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2answers
44 views

How can I solve this using the Lagrange method?

This is what I keep doing but the answer seems to be wrong every time.
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1answer
37 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 $g(x,y)=x^2-y^2-2$. ...
0
votes
2answers
148 views

Lagrange multipliers - finding maximum/minimum

I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and ...
0
votes
1answer
45 views

Constrained maximization of Leontif utility function $\min(x_1, x_2)$

The maximization problem is: Maximize $u(x_1, x_2) = \min[a_1x_1, a_2x_2]\; \ \text{s.t.}\;\; p_1x_1 + p_2x_2 \leq$ $w$, in which $x_i, p_i$ is the amount and price of good $i$, $w$ is the ...
0
votes
2answers
85 views

$f$ does not have extrema at the Lagrange multiplier value

I have the function $2x^2+y$ and one constraint $x-y^2=1$ and want to find maximum value by lagrange multiplier. Intuitively, I see the point $(2,1)$ satisfies $c$ and have value of $f(2,1)=9$. ...
3
votes
1answer
271 views

Largest box fitting inside an ellipsoid

Find the volume of the largest box with sides parallel to the $xy$, $xz$, and $yz$ planes that can fit inside the ellipsoid $(x/a)^2 + (y/b)^2 + (z/c)^2 = 1$. My answer: We want to maximize $f(x,y,z) ...
3
votes
1answer
434 views

Calculus of variations: Lagrange multipliers

Given a functional $$J(y)=\int_a^b F(x,y,y')dx, \tag{1}$$ where $y$ is a function of $x$, and a constraint $$\int_a^b K(x,y,y')dx=l, \tag{2}$$ if $y=y(x)$ is an extreme of (1) under the ...
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vote
1answer
143 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 ...
1
vote
1answer
219 views

Find max/min points for multivariable functions

I have a question about the general procedures to find the max/min points for multivariable functions, would really help if somebody could please clarify my doubts. So for single variable function, ...
2
votes
1answer
269 views

How to restrict Lagrange multiplier on positive values?

Here's the function that i want to optimize: $$f(x,y) = x-2y$$ and the constraint is: $$g(x,y) = x^2 + y - 10 = 0$$ Solving with Lagrange multiplier I get: $$F(x,y) = x-2y - x^2\lambda - y\lambda ...
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vote
2answers
1k 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 ...
3
votes
1answer
69 views

Can this system of equations be solved?

I'm doing some Lagrange multiplier examples that I thought up and I was trying to think of an example where the method of 'solving the constraint explicitly' fails. I'm trying to maximize ...
1
vote
1answer
54 views

Lagrange Multipliers restriction equation problem

There's a straight line going from B to C in the first quadrant of x,y coordinate system. B is $(0,s)$, C is $(t,0)$ Let A $(3,3)$ be a point on the line going from A to B. Find the equation of ...
0
votes
1answer
100 views

Find minima and maxima to the next function: $f(x,y)=xy$ in a blocked domain.

I've got the next function: $f(x,y)=xy$ I need to find the local minima / maxima in a block domain by $y=0$ and $y=x^2-4$. Largrange multipilers jumped to my mind. But i don't know how to choose a ...
0
votes
1answer
106 views

A hard multivariate optimization problem in $n-1$ variables

For $n>1$, I want to find the smallest value, and corresponding $x_i$ values, of $f(x_2,\dots,x_n) = \prod_{k=2}^n (x_k+1)^k$ subject to the constraints $x_j > 0$ for all $j$ and $\prod_{k=2}^n ...
0
votes
1answer
159 views

Right Triangles and Lagrange Multipliers

Suppose that you have a right triangle $a^2+b^2=c^2$ with integral sides. Given a perimeter $p=a+b+c$, how can you use Lagrange multipliers to determine the maximum length of $a$?
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vote
0answers
102 views

What do I need to know to understand Lagrange multipliers?

I've seen Lagrange multipliers used as a powerful method for tackling inequalities and some IMO problems, and I'm aware that it's a part of calculus. I'm currently taking BC Calculus in high school, ...
2
votes
4answers
384 views

Minimize $\sum a_i^2 \sigma^2$ subject to $\sum a_i = 1$

$$\min_{a_i} \sum_{i=1}^{n} {a_i}^2 \sigma^2\text{ such that }\sum_{i=1}^{n}a_i=1$$ and $\sigma^2$ is a scalar. The answer is $a_i=\frac{1}{n}$. I tried Lagrangian method. How can I get that ...
1
vote
1answer
120 views

HELP please with Optimization with constrain using lagrangian

I am reading this book on optimization and they present the following problem: Lisa wants to maximize her utility U(q1,q2) subject to a budget constrain, budget constrain is $p1*q1+p2*q2=I$. Ok , I ...
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vote
1answer
11k views

Deriving demand functions given utility

A consumer purchases food $X$ and clothing $Y$. Her utility function is given by: $U(X,Y) = XY +10Y$, income is $\$100$ the price of food is $\$1$ and the price of clothing is $P_y$. Derive the ...
1
vote
4answers
740 views

Use Lagrange multiplier to find absolute maximum and minimum

Use Lagrange multiplier to find absolute maximum and minimum of $f(x,y) =x^2+xy+y^2, x^2+y^2 =8$. What i've done so far.. $f_x = \lambda g_x \Rightarrow 2x+y =\lambda2x, \\f_y = \lambda g_y ...