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Home | 18.013A | Chapter 14 |
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Suppose we have a curve, that is defined by an equation, , and we seek an extremal value of among points restricted to lie on this curve.
Imagine, for example that represents an ellipse, , and we want the maximum of on .
At any point on , we are free to move while staying on only in the direction of the tangent line to the curve. Our condition above for an extremum then tells us that for to be an extremum of must have 0 derivative in the direction of the tangent, , to the curve defined by .
This means that the gradient of must be perpendicular to . But the gradient of is perpendicular to as well, so that in two dimensions the gradient of and the gradient of must be parallel, for to have an extremum on .
There are two standard ways to express this condition.
One is to notice that it means that the parallelogram formed by and has no area, so that the determinant with these vectors as columns must be 0.
The other is to notice that it means that for some constant .
Either observation allows us to find the extrema.
The second method is called that of "Lagrange Multipliers", and the constant is called a Lagrange Multiplier.
In the example you can click on above, if you write out the three equations defined by and , you may solve them for and , and arrive at the same solutions obtained.
Again, computing second derivatives (or examining values of ) must be used to determine the local and/or global maxima and minima.
When a curve is defined parametrically with parameter you can write and apply the single variable condition that .
Exercises:
14.1 Work out the details of the Lagrange Multipliers approach to the example above.
14.2 Suppose we want to maximize the volume of a vertically oriented cylinder given a fixed value for the surface area of its sides and its top (but not its bottom). What radius and height should it have?
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