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August 31st, 2013, 04:06 AM   #1
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minimum value of function

minimum value of
when subjected to and

satisfies the condition.
but I'm not able to show all three at a time
I want someone to explain me the problem.
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August 31st, 2013, 05:46 AM   #2
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Re: minimum value of function

Because of the cyclic symmetry of the variables, the minimum is found when:



You will easily find this is true using Lagrange multipliers.
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August 31st, 2013, 08:11 AM   #3
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Re: minimum value of function

I can see , and has similar role, nothing happens if we interchange their position.

could u please explain me in simple words.
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August 31st, 2013, 08:46 AM   #4
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Re: minimum value of function

Quote:
Originally Posted by MATHEMATICIAN
minimum value of
when subjected to and

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 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?
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August 31st, 2013, 09:42 AM   #5
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Re: minimum value of function

what I did is,

f = x + y + z . . . . (i)

x + y + z - 1 = 0 . . . . (ii)

xyz + 1 = 0 . . . . (iii)

we know,
x + y + z = ( x + y + z) - 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 = - 1 . . . . (v)

xy = - 1 . . . . (vi)

combining (v) and (vi)
yx - xy = 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 ?
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August 31st, 2013, 09:47 AM   #6
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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.
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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:











For which we find the real solutions:



and:

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August 31st, 2013, 10:21 AM   #8
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Re: minimum value of function

Could u please tell me someone about first three equations.

also, mew and lamda.
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August 31st, 2013, 10:27 AM   #9
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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 here to attempt a tutorial on the subject.
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August 31st, 2013, 10:33 AM   #10
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Re: minimum value of function

:'(
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