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inequalities (1 Viewer)

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Find the minimum value of x^2+y^2 given that xy(x^2-y^2)=x^2+y^2 x not = 0
 

cossine

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Find the minimum value of x^2+y^2 given that xy(x^2-y^2)=x^2+y^2 x not = 0
So you have not written coherently.

I am not 100% sure however I think Karush-Kuhn-Tucker condition is what you are after given you have additional constraint x != 0.

Not sure how to solve it otherwise.
 

Lith_30

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I have a slight feeling that this could have something to do with complex numbers, cause if you let

, and

so the equation can be or

I'm not sure where to go on from there, but maybe it could help you.

EDIT

maybe you can say that where

That is

Then
 
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cossine

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I have a slight feeling that this could have something to do with complex numbers, cause if you let

, and

so the equation can be or

I'm not sure where to go on from there, but maybe it could help you.

EDIT

maybe you can say that where

That is

Then
Sorry how did you get this part:

where
 

Lith_30

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Sorry how did you get this part:

where
It was an assumption that if was equivalent to the the imaginary component of then you could say for some real number a.

I just made the number equal zero cause I wanted the smallest value for .

Looking at it now, this is probably wrong. Cause there is no definite reason why has to be purely imaginary.
 

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i used trig sub and i got 1/2 sin(4theta)=1
 

Trebla

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A bit beyond syllabus but consider a parametric polar form where both r and theta are variables:



This reduces the condition to


Your objective is to find the minimum value of .

Since the RHS is fixed, we minimise by maximising .

Hence the minimum value of of is 4.
 

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