Method of Lagrange Multipliers for Solving Constrained Optimization Problems Calculus IV Lab Karen Donnelly Saint Joseph's College

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiQyQtSSV3aXRoRzYiNiNJKnBsb3R0b29sc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJSEiIg==PkktX0VudkV4cGxpY2l0RzYiSSV0cnVlRyUqcHJvdGVjdGVkRw== Note: the _EnvExplicit := true: option tells Maple to show all roots to equations explicitly. Load the Student[MultivariateCalculus] package to use the LagrangeMultipliers command. QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMmSShTdHVkZW50R0YnNiNJNU11bHRpdmFyaWF0ZUNhbGN1bHVzR0YoISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Calculate the maximum and minimum values of 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 subject to the constraint 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 Define the function f to maximize: PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmKiQ5JCIiIyIiIiokOSVGL0YwRiRGJEYk Define the constraint function g: PkkiZ0c2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwsKiQ5JCIiIyIiIiokOSVGL0YwRi5GL0YyISIjRjBGMEYkRiRGJA== Plot f and the constraint function g together and observe their points of intersection: QyQ+SSNnMUc2Ii1JL2ltcGxpY2l0cGxvdDNkR0YlNiovLUkiZ0dGJTYkSSJ4R0YlSSJ5R0YlIiIhL0YtOyEiIyIiIy9GLkYxL0kiekdGJTshIiUiIikvSSZzdHlsZUdGJUkscGF0Y2hub2dyaWRHRiUvSSZjb2xvckdGJUkkcmVkR0YlL0klZ3JpZEdGJTclIiM/RkNGQy9JLXRyYW5zcGFyZW5jeUdGJSQiIiYhIiJGSA== QyQ+SSNmMUc2Ii1JJ3Bsb3QzZEdGJTYoLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YsOyEiIyIiIy9GLUYvL0kmc3R5bGVHRiVJLXBhdGNoY29udG91ckdGJS9JKWNvbnRvdXJzR0YlIiNdL0ktdHJhbnNwYXJlbmN5R0YlJCIiJiEiIkY9 LUkoZGlzcGxheUc2IjYkPCRJI2YxR0YkSSNnMUdGJC9JJWF4ZXNHRiRJJmJveGVkR0Yk QyQ+SSJGRzYiLUkqdHJhbnNmb3JtR0YlNiNmKjYkSSJ4R0YlSSJ5R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiU3JTkkOSUtSSJmR0YlNiRGMUYyRiVGJUYlISIi PiZJInBHNiI2IyIiIi1JLWltcGxpY2l0cGxvdEdGJTYnLy1JImdHRiU2JEkieEdGJUkieUdGJSIiIS9GLzshIiMiIiMvRjBGMy9JJmNvbG9yR0YlSSRyZWRHRiUvSSp0aGlja25lc3NHRiUiIiQ= LUkoZGlzcGxheUc2IjYkNyRJI2YxR0YkLUkiRkdGJDYjJkkicEdGJDYjIiIiL0klYXhlc0dGJEkkYm94R0Yk Try to approximate where the maximum and minimum occur for this constrained problem from the graph above. Note that the maximum and mininimum will occur where a level curve of f(x,y) is tangent to the level curve g(x,y) = 0. To help visualize this, manipulate the graph above so that your perspective is from above looking down towards the x-y plane. We might be able to see this more clearly by a contour plot of the function f(x,y) in the plane along with the level curve g(x,y) = 0: QyQ+SSNwMUc2Ii1JLWltcGxpY2l0cGxvdEdGJTYnLy1JImdHRiU2JEkieEdGJUkieUdGJSIiIS9GLTshIiMiIiMvRi5GMS9JJWdyaWRHRiU3JCIjdkY4L0kmY29sb3JHRiVJJWJsdWVHRiUiIiI=QyQ+SSNwMkc2Ii1JLGNvbnRvdXJwbG90R0YlNiYtSSJmR0YlNiRJInhHRiVJInlHRiUvRiw7JCEjQSEiIiIiIi9GLTtGMiQiI0FGMi9JKWNvbnRvdXJzR0YlIiNdRjM=LUkoZGlzcGxheUc2IjYjPCRJI3AxR0YkSSNwMkdGJA== Since we know the gradient vectors for a function are normal to the level curves, we will have the gradient vectors for f and g are parallel at these points of maximum and minimum. Two vectors are parallel if and only if one is a scalar multiple of each the other -- the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJLyUrYmFja2dyb3VuZEdRKFswLDAsMF1GJy8lMGZvbnRfc3R5bGVfbmFtZUdRKl9jc3R5bGUxOEYnRjI= is this scalar. Thus we must solve equations 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 , 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 Applying the Method of Lagrange Multipliers: Form 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 PkkiaEc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmLUkiZkdGJDYkOSQ5JSIiIiomSSdsYW1iZGFHRiRGMi1JImdHRiRGL0YyISIiRiRGJEYk Differentiate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUShbMCwwLDBdRicvJTBmb250X3N0eWxlX25hbWVHUSpfY3N0eWxlMThGJy9GM1Enbm9ybWFsRic= with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2JhY2tncm91bmRHUShbMCwwLDBdRicvJTBmb250X3N0eWxlX25hbWVHUSpfY3N0eWxlMThGJy9GM1Enbm9ybWFsRic= and then with respect to y. PkkkZGh4RzYiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JImhHRiQ2JEkieEdGJEkieUdGJEYs PkkkZGh5RzYiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JImhHRiQ2JEkieEdGJEkieUdGJEYt The values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY5 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= which give the conditional maxima or minima of 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 are found by solving three equations: 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 Or with our shorthand function 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 QyQ+SSVzb2xuRzYiLUkmc29sdmVHRiU2JDwlL0kkZGh4R0YlIiIhL0kkZGh5R0YlRiwvLUkiZ0dGJTYkSSJ4R0YlSSJ5R0YlRiw8JUknbGFtYmRhR0YlRjNGNCIiIg==LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI= QyUmSSVzb2xuRzYiNiQiIiIiIiNGJyZGJDYkRiciIiQ= Check value of f(x,y) at the first of these two points by using the first result output above. QyQtSSJmRzYiNiQsJiEiIiIiIiokIiIjI0YpRisjRihGKywmRilGKUYqRixGKQ==LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI= And for the second: QyQtSSJmRzYiNiQsJiEiIiIiIiokIiIjI0YpRitGLCwmRilGKUYqI0YoRitGKQ==LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI= Since these are the extrema, clearly the first is a maximum and the second is a minimum. Check this result graphically: Plot the level curve of 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 corresponding to maximum value of f (5.828) and the curve representing g(x,y) = 0, which is the unit circle centred at (-1,1). They should be tangent. QyQ+SSVmbWF4RzYiJCIrQ3JVR2UhIioiIiI=PkklZm1pbkc2IiQiK2IoR2RyIiEjNQ== PkkmcGxvdDFHNiItSS1pbXBsaWNpdHBsb3RHRiQ2Ji8tSSJmR0YkNiRJInhHRiRJInlHRiRJJWZtYXhHRiQvRiw7ISIkIiIkL0YtRjAvSSZjb2xvckdGJEklQkxVRUdGJA== PkkmcGxvdDNHNiItSS1pbXBsaWNpdHBsb3RHRiQ2Ji8tSSJnR0YkNiRJInhHRiRJInlHRiQiIiEvRiw7ISIkIiIkL0YtRjAvSSZjb2xvckdGJEkkUkVER0Yk LUkoZGlzcGxheUc2IjYlPCRJJnBsb3QxR0YkSSZwbG90M0dGJC9JJnRpdGxlR0YkSTRDT05TVFJBSU5FRH5NQVhJTVVNR0YkL0koc2NhbGluZ0dGJEksY29uc3RyYWluZWRHRiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= PkkmcGxvdDJHNiItSS1pbXBsaWNpdHBsb3RHRiQ2Jy8tSSJmR0YkNiRJInhHRiRJInlHRiRJJWZtaW5HRiQvRiw7ISIkIiIkL0YtRjAvSSZjb2xvckdGJEkmR1JFRU5HRiQvSSVncmlkR0YkNyQiI11GOg== LUkoZGlzcGxheUc2IjYlPCRJJnBsb3QyR0YkSSZwbG90M0dGJC9JJnRpdGxlR0YkSTRDT05TVFJBSU5FRH5NSU5JTVVNR0YkL0koc2NhbGluZ0dGJEksY29uc3RyYWluZWRHRiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUkoZGlzcGxheUc2IjYlPCVJJnBsb3QxR0YkSSZwbG90MkdGJEkmcGxvdDNHRiQvSSZ0aXRsZUdGJEk4Q09OU1RSQUlORUR+TUFYfmFuZH5NSU5HRiQvSShzY2FsaW5nR0YkSSxjb25zdHJhaW5lZEdGJA== Note these values are not the maximum or minimum occurring values of 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, but the maximum and minimum values of f which lie on the circle 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 = 0. Note also the Lagrange multiplier LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpJmxhbWJkYTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrYmFja2dyb3VuZEdRKFswLDAsMF1GJy8lMGZvbnRfc3R5bGVfbmFtZUdRKl9jc3R5bGUxOEYnRjI= need not be evaluated. The Student[MultivariateCalculus] package has a built-in function that will calculate the extremum of a function subject to a constraint for us -- we check our answer. Note we can add various options to this command: LUk0TGFncmFuZ2VNdWx0aXBsaWVyc0c2IjYlLCYqJEkieEdGJCIiIyIiIiokSSJ5R0YkRilGKjcjLCxGJ0YqRitGKkYoRilGLCEiI0YqRio3JEYoRiw= LUk0TGFncmFuZ2VNdWx0aXBsaWVyc0c2IjYmLCYqJEkieEdGJCIiIyIiIiokSSJ5R0YkRilGKjcjLCxGJ0YqRitGKkYoRilGLCEiI0YqRio3JEYoRiwvSSdvdXRwdXRHRiRJKWRldGFpbGVkR0Yk LUk0TGFncmFuZ2VNdWx0aXBsaWVyc0c2IjYnKiZJInlHRiQiIiJJInhHRiRGKDcjLCgqJEYpIiIjI0YoIiIpKiRGJ0YtI0YoRi0hIiJGKDckRilGJy9JJ291dHB1dEdGJEklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJC9JMHNob3dsZXZlbGN1cnZlc0dGJEkmZmFsc2VHRjg=

Using the same procedures, calculate the maximum and minimum values of: 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 subject to the constraint 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 Before we begin ask Maple to find values that we can check our work against: LUk0TGFncmFuZ2VNdWx0aXBsaWVyc0c2IjYlLCgqJEkieEdGJCIiIyIiKSomSSJ5R0YkIiIiRihGLSEjNyokRixGKSIjPDcjLChGJ0YtRi9GLSEiIkYtNyRGKEYs Define function f and constraint function g: PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoKiQ5JCIiIyIiKSomOSUiIiJGLkYzISM3KiRGMkYvIiM8RiRGJEYk PkkiZ0c2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoKiQ5JCIiIyIiIiokOSVGL0YwISIiRjBGJEYkRiQ= QyQ+SSNnMUc2Ii1JL2ltcGxpY2l0cGxvdDNkR0YlNicvLCgqJEkieEdGJSIiIyIiIiokSSJ5R0YlRi1GLiEiIkYuIiIhL0YsO0YxRi4vRjBGNC9JInpHRiU7ISIlIiQrIi9JJmNvbG9yR0YlSSRyZWRHRiVGMQ== QyQ+SSNmMUc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYoLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YvOyQhI0QhIiIkIiNERjUvRjBGMi9JJnN0eWxlR0YlSS1wYXRjaGNvbnRvdXJHRiUvSS10cmFuc3BhcmVuY3lHRiUkIiImRjUvSSljb250b3Vyc0dGJSIjU0Y1 LUkoZGlzcGxheUc2IjYkPCRJI2YxR0YkSSNnMUdGJC9JJWF4ZXNHRiRJJmJveGVkR0Yk PkkiaEc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmLUkiZkdGJDYkOSQ5JSIiIiomSSdsYW1iZGFHRiRGMi1JImdHRiRGL0YyISIiRiRGJEYk Differentiate h with respect to x and then with respect to y. PkkkZGh4RzYiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JImhHRiQ2JEkieEdGJEkieUdGJEYs PkkkZGh5RzYiLUklZGlmZkclKnByb3RlY3RlZEc2JC1JImhHRiQ2JEkieEdGJEkieUdGJEYt Solve these three equations Pkklc29sbkc2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQ8JS9JJGRoeEdGJCIiIS9JJGRoeUdGJEYuLy1JImdHRiQ2JEkieEdGJEkieUdGJEYuPCVJJ2xhbWJkYUdGJEY1RjY= There are two constrained maxima and two constrained minima. PkklZm1pbkc2Ii1JImZHRiQ2JC1JIipHJSpwcm90ZWN0ZWRHNiQsJCokIiImIyIiIiIiI0YxI0YwRi4sJEYtRjI= PkklZm1heEc2Ii1JImZHRiQ2JCwkKiQiIiYjIiIiIiIjI0YsRiosJC1JIipHJSpwcm90ZWN0ZWRHNiQsJEYpRi1GLiEiIg== Now plot the functions to confirm these answers - plot the constraint in red. Plot the level curve for f corresponding to the maximum and minimums. PkkmcGxvdDFHNiItSS1pbXBsaWNpdHBsb3RHRiQ2Ji8tSSJmR0YkNiRJInhHRiRJInlHRiRJJWZtYXhHRiQvRiw7ISIjIiIjL0YtRjAvSSZjb2xvckdGJEklQkxVRUdGJA== PkkmcGxvdDNHNiItSS1pbXBsaWNpdHBsb3RHRiQ2Ji8tSSJnR0YkNiRJInhHRiRJInlHRiQiIiEvRiw7ISIjIiIjL0YtRjAvSSZjb2xvckdGJEkkUkVER0Yk LUkoZGlzcGxheUc2IjYlPCVJJnBsb3QxR0YkSSZwbG90MkdGJEkmcGxvdDNHRiQvSShzY2FsaW5nR0YkSSxjb25zdHJhaW5lZEdGJC9JJnRpdGxlR0YkSThDT05TVFJBSU5FRH5NQVh+YW5kfk1JTkdGJA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

solve From plots implictplot implictplot3d contourplot From plottools transform From Student[MultivariateCalculus] LagrangeMultipliers