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 August 31st, 2013, 04:06 AM #1 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित minimum value of function minimum value of $x^2 + y^2 + z^2$ when subjected to $x + y + z -1= 0$ and $xyz + 1= 0$ $(1, 1, -1), (1, -1,1) & (-1, 1, 1)$ satisfies the condition. but I'm not able to show all three at a time I want someone to explain me the problem.
 August 31st, 2013, 05:46 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: minimum value of function Because of the cyclic symmetry of the variables, the minimum is found when: $x=y=z=\frac{1}{3}$ You will easily find this is true using Lagrange multipliers.
 August 31st, 2013, 08:11 AM #3 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित Re: minimum value of function I can see $x$, $y$ and $z$ has similar role, nothing happens if we interchange their position. could u please explain me in simple words.
August 31st, 2013, 08:46 AM   #4
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Re: minimum value of function

Quote:
 Originally Posted by MATHEMATICIAN minimum value of $x^2 + y^2 + z^2$ when subjected to $x + y + z -1= 0$ and $xyz + 1= 0$ $(1, 1, -1), (1, -1,1) & (-1, 1, 1)$ satisfies the condition. but I'm not able to show all three at a time I want someone to explain me the problem.
I don't understand what you mean by "show all three at a time". I presume that "all three" refers to minimizing $x^2+ y^2= z^2$ while satisfying x+ y+ z= 1 and xyz= -1. But what do you mean by "at a time"?

In any case, what have you tried?

 August 31st, 2013, 09:42 AM #5 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित Re: minimum value of function what I did is, f = x$^2$ + y$^2$ + z$^2$ . . . . (i) x + y + z - 1 = 0 . . . . (ii) xyz + 1 = 0 . . . . (iii) we know, x$^2$ + y$^2$ + z$^2$ = ( x + y + z)$^2$ - 2xy - 2yz - 2xz f = 1 - 2xy - 2yz - 2xz substituting value of "z" from equation (iii) f = 1 - 2xy + 2/x + 2/y . . . . (vi) partially differentiating equation (vi) w.r.t x and y and equation the results with zero, yx$^2$ = - 1 . . . . (v) xy$^2$ = - 1 . . . . (vi) combining (v) and (vi) yx$^2$ - xy$^2$ = 0 xy ( x - y ) = 0 thus, x = y why am i getting x = y if x = y, I get only (1, 1, -1) and don't get other point. is the process I followed incorrect ?
 August 31st, 2013, 09:47 AM #6 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित Re: minimum value of function [color=#800040]Mark[/color] there are two conditions, x + y + z - 1 = 0 and xyz + 1 = 0 I don't know latex properly, so may b second condition is not displayed properly.
August 31st, 2013, 10:08 AM   #7
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Re: minimum value of function

Quote:
 Originally Posted by MATHEMATICIAN [color=#800040]Mark[/color] there are two conditions, x + y + z - 1 = 0 and xyz + 1 = 0 I don't know latex properly, so may b second condition is not displayed properly.
Yes, I see that now after [color=#00BF00]HallsofIvy [/color]'s post. In your original post this was not clear. Lagrange multipliers here implies the system:

$2x=\lambda+\mu yz$

$2y=\lambda+\mu xz$

$2z=\lambda+\mu xy$

$x+y+z-1=0$

$xyz\,+\,1\,=\,0$

For which we find the real solutions:

$(\lambda,\mu)=(0,-2)$

and:

$(x,y,z)=(-1,1,1),\,(1,-1,1),\,(1,1,-1)$

 August 31st, 2013, 10:21 AM #8 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित Re: minimum value of function Could u please tell me someone about first three equations. also, mew and lamda.
 August 31st, 2013, 10:27 AM #9 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: minimum value of function I assumed that if you are studying optimization with constraints, then you are familiar with Lagrange multipliers. I simply do not have the patience with the quirky rendering of $\LaTeX$ here to attempt a tutorial on the subject.
 August 31st, 2013, 10:33 AM #10 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित Re: minimum value of function :'(

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