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How to Read and Do Proofs: An Introduction to Mathematical Thought Processes |
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. |
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 |
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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) | |
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 2) | |
Two Constraints |
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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 | |
Equations That Do Not Use The Gradient |
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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 |
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Practice Problems |
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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.
Level A - Basic |
Practice A01 | |
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optimize: \( f(x,y) = x^2+y^2 \) | |
solution |
Practice A02 | |
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optimize: \( f(x,y) = x^2+y^2 \) | |
solution |
Practice A03 | |
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minimize: \( C(x,y) = 6x^2 + 12y^2 \) | |
solution |
Practice A04 | |
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optimize: \( f(x,y) = xy \) | |
solution |
Practice A05 | |
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minimize: \( f(x,y,z) = x^2+y^2+z^2 \) | |
solution |
Practice A06 | |
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maximize: \( f(x,y,z) = x^2+2y-z^2 \) | |
solution |
Practice A07 | |
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optimize: \( f(x,y) = 3x+4y \) | |
solution |
Practice A08 | |
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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 |
Practice A09 | |
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maximize: \( T(x,y) = 6xy \) | |
solution |
Practice A10 | |
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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 |
Practice A11 | |
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Find the rectangle with maximum perimeter that can inscribed in the ellipse \(x^2+4y^2 = 4\). | |
solution |
Practice A12 | |
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optimize: \( f(x,y,z)=xyz \) | |
solution |
Practice A13 | |
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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 |
Level B - Intermediate |
Practice B01 | |
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optimize: \( f(x,y) = (x-1)^2 + (y-2)^2 - 4 \) | |
solution |
Practice B02 | |
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optimize: \( f(x,y,z) = x^2+y^2+z^2 \) | |
solution |
Practice B03 | |
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optimize: \( f(x,y,z) = 3x-y-3z \) | |
solution |
Practice B04 | |
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optimize: \( f(x,y,z) = x^2+x+2y^2+3z^2 \) | |
solution |