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Find three numbers whose sum is 12 if their product is to be as large as possible.

$$(4,4,4)$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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02:19

Find three positive number…

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03:20

Find two numbers whose sum…

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01:59

Find numbers $x$ and $y$ s…

for this problem we are asked to find three numbers whose sum is 12 if the product is to be as large as possible. So this is a constrained optimization problem. Our function that we want to optimize, which I'm going to call F of X, Y. Z is just equal to X times Y times ed. And our constraint is that G of X, Y. Z equals the sum, so X plus Y plus said. And it must equal 12. So as a constrained optimization problem, we can solve this using the method of lagrange multipliers, but the gradient of F two equal lambda, temp gradient of G. And we'll have a system of four equations with foreign knowns that we can solve. So the first equation that will get is that why I said must equal lambda. Then the second equation is that exit must equal lambda. And the third equation is that X. Y must equal lambda which tells us by extension that xy must equal exit must equal wise it so that tells us that X plus Y plus Z Must equal then one moment here, X plus Y plus said then must actually equal three times route lambda. Because from that first set of three equations, we'll get that X equals Y equals Ed. And whatever they have to be, each one must equal the square root of lambda. So we have that X plus Y plus. It must equal three route lambda route lambda which equals 12, which then means that route lambda must equal 12/3. So that has to equal four, which means that lambda equals 16. So we get then that X equals Y equals Ed, and they all equal four.

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