-1
votes
0answers
23 views

Constraint optimization with lagrangian

I am having trouble regarding the general steps one needs to take in order to solve an constraint optimization using Lagrangian. More specifically, I want to maximize objective equation $f(x,y,z,w)$ ...
0
votes
2answers
22 views

Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
3
votes
2answers
146 views

Maximizing Area of Triangle in Circle

I was playing around with another example that I made up where I am trying to maximize the area of a triangle inscribed in a circle of radius. I want to do the problem using the method of Lagrange ...
1
vote
2answers
26 views

Tips on resolving a Lagrange Multipliers equation system

I'm having a very hard time resolving the system of equations after using the Lagrange Multipliers optimization method. For instance: The plane $ x + y + 2z = 2 $ intersects the paraboloid $ z = x^2 ...
-1
votes
0answers
24 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?
1
vote
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 ...
0
votes
1answer
40 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
1
vote
1answer
22 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 ...
0
votes
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 ...
0
votes
3answers
41 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
1
vote
1answer
39 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 ...
2
votes
2answers
26 views

In regards to lagrange multipliers, Confusion about derivation.

In my calculus III textbook, the following sentence is causing trouble for me and preventing me from understanding the theory behind Lagrange multipliers. "Since the gradient vector for a given ...
0
votes
1answer
60 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
37 views

Use Lagrange Multipliers to determine max and min

Using Lagrange Multipliers, determine the maximum and minimum of the function $f(x,y,z) = x + 2y$ subject to the constraints $x + y + z = 1$ and $y^2 + z^2 = 4$: Justify that the points you have found ...
0
votes
1answer
46 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
4
votes
0answers
45 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 ...
1
vote
1answer
76 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
vote
1answer
49 views

Maximum distance from the origin to the surface

I am having trouble getting the maximum distance from the origin to the surface $$ \frac{x^4}{16} +\frac{y^4}{81} + z^4 = 1 $$ Knowing I have to maximize $x^2 +y^2+ z^2$ and that the constrain ...
1
vote
1answer
61 views

Finding the shortest distance between two planes using Lagrange multipliers

A problem (among a list of Lagrange multipliers problems in Earl Swokowski's Calculus) states as follows: find the shortest distance between $2x+3y-z = 2$ and $2x+3y-z=4$. I can see that the ...
2
votes
3answers
57 views

Extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange Multipliers

Find the extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange multipliers. So I set it up: $$ 1 = 2x\lambda_1 + 2\lambda_2 \\ 1 = -2y\lambda_1 \\ 1 = \lambda_2 $$ Plug ...
0
votes
1answer
36 views

Finding critical points using Lagrange multipliers

So I got a bit stuck with the technicalities of this exercise. Let $f(x,y,z)=\sqrt {x^2+y^2+z^2}$. Find $f$ minimal points under the constraints $x^2+y^2=1$ and $x^2-xy+y^2-z^2=1$ using Lagrange ...
2
votes
1answer
80 views

The meaning of $\lambda$ in Lagrange Multipliers

This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent. I am wondering if the ...
2
votes
3answers
56 views

Prove $\sin \frac{\alpha}{2}\sin \frac{\beta}{2}\sin \frac{\gamma}{2}\leq \frac {1}{8}$, $\alpha, \gamma\, \beta$ being angles of a triangle

Prove $\sin \frac{\alpha}{2}\sin \frac{\beta}{2}\sin \frac{\gamma}{2}\leq \frac {1}{8}$ I defined $f(x,y,z)=\sin \frac{\alpha}{2}\sin \frac{\beta}{2}\sin \frac{\gamma}{2}$, and wanted to find max/min ...
0
votes
2answers
82 views

Distance from Ellipsoid to Plane - Lagrange Multiplier

Find the distance from the ellipsoid $x^2 + y^2 + 4z^2 = 4$ to the plane $x + y + z = 6$. I'm trying to do it using Lagrange multipliers over the distance equation, but then it just gets ...
0
votes
1answer
28 views

using the method of LaGrange multipliers find the extreme values of the function

using the method of LaGrange multipliers find the extreme values of the function $f(x,y)=x^3+4y^2$ with constraint $x^2+y^2=1$ currently I have that: $3x^2=2Lx$ which leads to $x=0$ or $L=3/2$ and: ...
4
votes
2answers
67 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 ...
0
votes
1answer
28 views

Maximize/minimize $1/3 x^3 + y$ with constraint $x^2 + y^2 = 1$?

I keep running around in circles when I use the Lagrangian multiplier method getting $x = 1/y$ But then when I substitute $(1/y)^2 + y^2 = 1$ I then get $1/y^2 + y^2 = 1$ and this doesn't give me ...
1
vote
2answers
52 views

Using the method of Lagrange multipliers, find the extreme values of a function

Using the method of Lagrange multipliers, find the extreme values of the function $f(x,y)= \frac{2y^3}{3} + 2x^2 +1$ on the ellipse $5x^2 + y^2 = 1/9$ . Identify the (absolute) maximal and minimal ...
0
votes
1answer
79 views

Extreme value Lagrange multiplier (max or min?)

I am to determine the the range of the volume of a tetrahedron enclosed by the coordinate axes and a tangentplane on the ellipsoid $x^2 + 2y^2 + 3z^2 = 1$. The volume of the tetrahedron can be derived ...
1
vote
2answers
34 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
102 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
2answers
40 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
3answers
62 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
65 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
50 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
41 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
52 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
66 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
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 ...
2
votes
3answers
135 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
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 ...
1
vote
0answers
16 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
35 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
93 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
41 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
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 $g(x,y)=x^2-y^2-2$. ...
2
votes
0answers
116 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
121 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
40 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
47 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 ...