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 MATHEMATICIAN August 31st, 2013 04:06 AM

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.

 MarkFL August 31st, 2013 05:46 AM

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.

 MATHEMATICIAN August 31st, 2013 08:11 AM

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.

 HallsofIvy August 31st, 2013 08:46 AM

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?

 MATHEMATICIAN August 31st, 2013 09:42 AM

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 ?

 MATHEMATICIAN August 31st, 2013 09:47 AM

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.

 MarkFL August 31st, 2013 10:08 AM

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:

 MATHEMATICIAN August 31st, 2013 10:21 AM

Re: minimum value of function

Could u please tell me someone about first three equations.

also, mew and lamda.

 MarkFL August 31st, 2013 10:27 AM

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.

 MATHEMATICIAN August 31st, 2013 10:33 AM

Re: minimum value of function

:'(

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