GRE 8767# 36

Forum for the GRE subject test in mathematics.
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Joined: Mon Sep 29, 2014 2:45 am

GRE 8767# 36

Postby shalale » Mon Oct 20, 2014 2:56 am

Hi freinds;
can someone give me the solution of #36 on GRE 8767?
I know how I should solve it but I cannot get the answer.
thank you :)

Posts: 26
Joined: Fri Jun 01, 2012 9:32 am

Re: GRE 8767# 36

Postby Austin » Mon Oct 20, 2014 3:03 pm

We're trying to minimize the function d(x,y)=\sqrt{x^2+y^2}, subject to the restriction xy=8, which should immediately suggest the Lagrange multipliers method.

Of course, it's easier to work with D(x,y)=x^2+y^2. We can minimize this function, then take a square root. Then, letting g(x,y)=xy, we set \nabla D = \langle 2x,2y \rangle = \lambda\nabla g = \lambda\langle x,y \rangle, so we have 2x &= \lambda y,~~~~<br />2y &= \lambda x,~~~~<br />8 &= xy.

We can quickly work out that the first two equations are satisfied when either y=x or y=-x. But y=-x can never satisfy 8=xy, so we must have y=x, and thus 8=x^2 and 8=y^2, so D(x,y)=16, and d(x,y)=4.

This is the "this is a timed test and we're in a hurry" version. I'm sure someone else could give a more lucid explanation.

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