17Calculus - Lagrange Multipliers
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.
Recommended Books on Amazon | ||
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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 |
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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 |
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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 |
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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 |
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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 |
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 |
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optimize: \( f(x,y) = x^2+y^2 \) |
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constraint: \( 2x+6y=2000 \) |
Problem Statement |
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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 |
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This solution is in two consecutive videos.
812 video
video by Krista King Math |
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812 video
video by Krista King Math |
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optimize: \( f(x,y) = x^2+y^2 \) |
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constraint: \( xy=1 \) |
Problem Statement |
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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 |
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This solution is in two consecutive videos.
813 video
video by Krista King Math |
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813 video
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 |
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Minimize \( C(x,y) = 6x^2 + 12y^2 \) subject to the constraint \( x + y = 90 \).
Solution |
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816 video
video by PatrickJMT |
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optimize: \( f(x,y) = xy \) |
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constraint: \( x^2 + 2y^2 = 1 \) |
Problem Statement |
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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 |
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818 video
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 |
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Minimize \( f(x,y,z) = x^2 + y^2 + z^2 \) subject to the constraint \( 2x + y - z = 1 \).
Solution |
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819 video
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 |
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Maximize \( f(x,y,z) = x^2+2y-z^2 \) subject to the constraints \( 2x=y \) and \( y+z=0 \).
Solution |
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820 video
video by Dr Chris Tisdell |
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optimize: \( f(x,y) = 3x+4y \) |
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constraint: \( x^2+y^2 = 1 \) |
Problem Statement |
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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 |
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821 video
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 |
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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 \( z > 0 \).
Solution |
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822 video
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 |
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Maximize \( T(x,y) = 6xy \) subject to the constraint \( x^2 + y^2 = 8 \).
Solution |
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823 video
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 |
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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.
Solution |
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824 video
video by Dr Chris Tisdell |
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close solution
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optimize: \( f(x,y,z) = xyz \) |
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constraint: \( x^2 + y^2 + z^2 = 3 \) |
Problem Statement |
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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 |
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1825 video
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 |
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Find the maximum and minimum values of the function \( f(x,y) = e^{xy} \) subject to \( x^3 + y^3 = 16 \).
Solution |
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1924 video
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 |
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Find the rectangle with maximum perimeter that can inscribed in the ellipse \( x^2 + 4y^2 = 4 \).
Solution |
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825 video
video by MIT OCW |
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Intermediate |
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optimize: \( f(x,y) = (x-1)^2 + (y-2)^2 - 4 \) |
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constraint: \( 3x + 5y = 47 \) |
Problem Statement |
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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 |
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This solution is in three consecutive videos.
814 video
video by Krista King Math |
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814 video
video by Krista King Math |
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814 video
video by Krista King Math |
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optimize: \( f(x,y,z) = x^2+y^2+z^2 \) |
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constraints: \( x+y+z = 1 \) and \( x+2y+3z = 6 \) |
Problem Statement |
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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 |
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815 video
video by Krista King Math |
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close solution
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optimize: \( f(x,y,z) = 3x-y-3z \) |
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constraints: \( x+y-z=0 \) and \( x^2+2z^2 = 1 \) |
Problem Statement |
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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 |
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This solution is presented in two consecutive videos.
817 video
video by PatrickJMT |
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817 video
video by PatrickJMT |
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close solution
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optimize: \( f(x,y,z) = x^2+x+2y^2+3z^2 \) |
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constraint: \( x^2+y^2+z^2 = 1 \) |
Problem Statement |
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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 |
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826 video
video by MIT OCW |
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lagrange multipliers 17calculus youtube playlist
Here is a playlist of the videos on this page.
You CAN Ace Calculus
Topics You Need To Understand For This Page
Trig Formulas
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
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Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
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Set 2 - squared identities | ||
|---|---|---|
\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
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Set 3 - double-angle formulas | |
|---|---|
\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
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Set 4 - half-angle formulas | |
|---|---|
\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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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.
