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17Calculus - Lagrange Multipliers

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The Method of Lagrange Multipliers is used to find maximums and minimums of a function subject to one or more constraints. We could also say that we want to optimize the function or find the extreme values of the function.
We highly recommend that you download the notes for this topic from Dr Chris Tisdell. Look for the pdf link entitled Extreme values + Lagrange multipliers.

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In basic calculus, we learned that finding the critical points gives us information about maximums, minimums (and saddlepoints). We use the same idea here, i.e. locations where the derivative is zero gives us possible locations of maxs/mins. We use this method to integrate the constraints into the equation.

Basic Technique

If we are given a function \(f(x,y,z)\) that we want to optimize (find maximums, minimums or both) subject to a constraint \(g(x,y,z) = 0\), we set up the gradient equation \( \nabla f = \lambda \nabla g \). [ For a version of the equations that do not use the gradient, see below. ]

The variable \( \lambda \) is just a number (independent of \(x\), \(y\) and \(z\)) called a Lagrange multiplier. We introduce \( \lambda \) during the course of solving this problem but it will not appear in our answer.

Remember that the gradient is a vector. So, the above equation gives us three equations ( one for each variable ) and four unknowns ( \(x\), \(y\), \(z\) and \( \lambda \) ). Using also the constraint equation \(g(x,y,z) = 0\), we can now (theoretically) solve for all four unknowns.

This next video clip explains this technique in more detail.

Dr Chris Tisdell - Lagrange Multipliers (clip 1) [6mins-23secs]

video by Dr Chris Tisdell

So, why does this work? It seems kind of strange that introducing another variable would enable us to optimize a function. Here is a great video clip that explains this. His use of graphs is very good to visualize what is going on.

Dr Chris Tisdell - Lagrange Multipliers (clip 1) [8mins-17secs]

video by Dr Chris Tisdell

Two Constraints

Okay, so now you know how to handle one constraint, if you are given two constraints, you just add another lagrange multiplier. We usually use the Greek letter mu, \( \mu \). The equation then looks like \( \nabla f = \lambda \nabla g + \mu \nabla h \) where the function we want to optimize is \(f(x,y,z)\) and the constraint equations are \(g(x,y,z) = 0\) and \(h(x,y,z) = 0\). Here is a video explaining this in more detail, including an example.

Dr Chris Tisdell - Lagrange Multipliers: 2 Constraints [14mins-23secs]

video by Dr Chris Tisdell

Equations That Do Not Use The Gradient

An alternate version of the equations using the Lagrange Method that do not use the gradient is given below. We present equations with three variables and two constraints. As you would expect, you can use the same ideas with two variables (drop \(z\)) and one constraint (drop \( h(x,y,z) \)).

Equations And Set Up

Optimize \(f(x,y,z)\)

Constraints \(g(x,y,z)=0\) and \(h(x,y,z) = 0\)

\( L(x,y,z,\lambda, \mu) = f(x,y,z) - \lambda g(x,y,z) - \mu h(x,y,z) \)
 

Equations To Solve

\( \partial L / \partial x = 0 ~~~ \to ~~~ \partial f / \partial x - \lambda ~ \partial g / \partial x - \mu ~ \partial h / \partial x = 0 \)

\( \partial L / \partial y = 0 ~~~ \to ~~~ \partial f / \partial y - \lambda ~ \partial g / \partial y - \mu ~ \partial h / \partial y = 0 \)

\( \partial L / \partial z = 0 ~~~ \to ~~~ \partial f / \partial z - \lambda ~ \partial g / \partial z - \mu ~ \partial h / \partial z = 0 \)

\( g(x,y,z) = 0 \)

\( h(x,y,z) = 0 \)

5 equations and 5 unknowns

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Practice

Unless otherwise instructed, follow these guidelines.
1. Optimize the functions subject to the given constraints.
2. Show whether they are maximums or minimums.
3. If you are not told whether to find maximums or minimums, find all of them.
4. Give all answers in exact form.

Basic

optimize: \( f(x,y) = x^2+y^2 \)

constraint: \( 2x+6y=2000 \)

Problem Statement

1. Optimize the function \( f(x,y) = x^2+y^2 \) subject to the constraint \( 2x+6y=2000 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give all answers in exact form.

Solution

This solution is in two consecutive videos.

Krista King Math - 812 video solution

video by Krista King Math

Krista King Math - 812 video solution

video by Krista King Math

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optimize: \( f(x,y) = x^2+y^2 \)

constraint: \( xy=1 \)

Problem Statement

1. Optimize the function \( f(x,y) = x^2+y^2 \) subject to the constraint \( xy=1 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give your answers in exact form.

Solution

This solution is in two consecutive videos.

Krista King Math - 813 video solution

video by Krista King Math

Krista King Math - 813 video solution

video by Krista King Math

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Minimize \( C(x,y) = 6x^2 + 12y^2 \) subject to the constraint \( x + y = 90 \).

Problem Statement

Minimize \( C(x,y) = 6x^2 + 12y^2 \) subject to the constraint \( x + y = 90 \).

Solution

PatrickJMT - 816 video solution

video by PatrickJMT

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optimize: \( f(x,y) = xy \)

constraint: \( x^2 + 2y^2 = 1 \)

Problem Statement

1. Optimize the function \( f(x,y) = xy \) subject to the constraint \( x^2 + 2y^2 = 1 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give all answers in exact form.

Solution

Dr Chris Tisdell - 818 video solution

video by Dr Chris Tisdell

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Minimize \( f(x,y,z) = x^2 + y^2 + z^2 \) subject to the constraint \( 2x + y - z = 1 \).

Problem Statement

Minimize \( f(x,y,z) = x^2 + y^2 + z^2 \) subject to the constraint \( 2x + y - z = 1 \).

Solution

Dr Chris Tisdell - 819 video solution

video by Dr Chris Tisdell

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Maximize \( f(x,y,z) = x^2+2y-z^2 \) subject to the constraints \( 2x=y \) and \( y+z=0 \).

Problem Statement

Maximize \( f(x,y,z) = x^2+2y-z^2 \) subject to the constraints \( 2x=y \) and \( y+z=0 \).

Solution

Dr Chris Tisdell - 820 video solution

video by Dr Chris Tisdell

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optimize: \( f(x,y) = 3x+4y \)

constraint: \( x^2+y^2 = 1 \)

Problem Statement

1. Optimize the function \( f(x,y) = 3x+4y \) subject to the constraint \( x^2+y^2 = 1 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give all answers in exact form.

Solution

Dr Chris Tisdell - 821 video solution

video by Dr Chris Tisdell

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Determine the point on the surface of \( xyz = 1 \) that is closest to the origin and satifies \( x > 0 \), \( y > 0 \) and \( z > 0 \).

Problem Statement

Determine the point on the surface of \( xyz = 1 \) that is closest to the origin and satifies \( x > 0 \), \( y > 0 \) and \( z > 0 \).

Solution

Dr Chris Tisdell - 822 video solution

video by Dr Chris Tisdell

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Maximize \( T(x,y) = 6xy \) subject to the constraint \( x^2 + y^2 = 8 \).

Problem Statement

Maximize \( T(x,y) = 6xy \) subject to the constraint \( x^2 + y^2 = 8 \).

Solution

Dr Chris Tisdell - 823 video solution

video by Dr Chris Tisdell

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We have a thin metal plate that occupies the region in the xy-plane \( x^2 + y^2 \leq 25 \). If \( f(x,y) = 4x^2 - 4xy + y^2 \) denotes the temperature (in degrees C) at any point on the plate, determine the highest and lowest temperatures on the edge of the plate.

Problem Statement

We have a thin metal plate that occupies the region in the xy-plane \( x^2 + y^2 \leq 25 \). If \( f(x,y) = 4x^2 - 4xy + y^2 \) denotes the temperature (in degrees C) at any point on the plate, determine the highest and lowest temperatures on the edge of the plate.

Solution

Dr Chris Tisdell - 824 video solution

video by Dr Chris Tisdell

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optimize: \( f(x,y,z) = xyz \)

constraint: \( x^2 + y^2 + z^2 = 3 \)

Problem Statement

1. Optimize the function \( f(x,y,z) = xyz \) subject to the constraint \( x^2 + y^2 + z^2 = 3 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give all answers in exact form.

Solution

Krista King Math - 1825 video solution

video by Krista King Math

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Find the maximum and minimum values of the function \( f(x,y) = e^{xy} \) subject to \( x^3 + y^3 = 16 \).

Problem Statement

Find the maximum and minimum values of the function \( f(x,y) = e^{xy} \) subject to \( x^3 + y^3 = 16 \).

Solution

MIP4U - 1924 video solution

video by MIP4U

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Find the rectangle with maximum perimeter that can inscribed in the ellipse \( x^2 + 4y^2 = 4 \).

Problem Statement

Find the rectangle with maximum perimeter that can inscribed in the ellipse \( x^2 + 4y^2 = 4 \).

Solution

MIT OCW - 825 video solution

video by MIT OCW

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Intermediate

optimize: \( f(x,y) = (x-1)^2 + (y-2)^2 - 4 \)

constraint: \( 3x + 5y = 47 \)

Problem Statement

1. Optimize the function \( f(x,y) = (x-1)^2 + (y-2)^2 - 4 \) subject to the constraint \( 3x + 5y = 47 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give all answers in exact form.

Solution

This solution is in three consecutive videos.

Krista King Math - 814 video solution

video by Krista King Math

Krista King Math - 814 video solution

video by Krista King Math

Krista King Math - 814 video solution

video by Krista King Math

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optimize: \( f(x,y,z) = x^2+y^2+z^2 \)

constraints: \( x+y+z = 1 \) and \( x+2y+3z = 6 \)

Problem Statement

1. Optimize the function \( f(x,y,z) = x^2+y^2+z^2 \) subject to the constraints \( x+y+z = 1 \) and \( x+2y+3z = 6 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give all answers in exact form.

Solution

Krista King Math - 815 video solution

video by Krista King Math

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optimize: \( f(x,y,z) = 3x-y-3z \)

constraints: \( x+y-z=0 \) and \( x^2+2z^2 = 1 \)

Problem Statement

1. Optimize the function \( f(x,y,z) = 3x-y-3z \) subject to the constraints \( x+y-z=0 \) and \( x^2+2z^2 = 1 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give all answers in exact form.

Solution

This solution is presented in two consecutive videos.

PatrickJMT - 817 video solution

video by PatrickJMT

PatrickJMT - 817 video solution

video by PatrickJMT

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optimize: \( f(x,y,z) = x^2+x+2y^2+3z^2 \)

constraint: \( x^2+y^2+z^2 = 1 \)

Problem Statement

1. Optimize the function \( f(x,y,z) = x^2+x+2y^2+3z^2 \) subject to the constraint \( x^2+y^2+z^2 = 1 \).
2. Find all of the maximums and minimums and show whether they are maximums or minimums.
3. Give all answers in exact form.

Solution

MIT OCW - 826 video solution

video by MIT OCW

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Basic Technique

Two Constraints

Equations That Do Not Use The Gradient

Practice

Page Sections

Practice Instructions

Unless otherwise instructed, follow these guidelines.
1. Optimize the functions subject to the given constraints.
2. Show whether they are maximums or minimums.
3. If you are not told whether to find maximums or minimums, find all of them.
4. Give all answers in exact form.

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