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For any x,y, in R,xygt0 . Then the minim...

For any `x,y, in R,xygt0` . Then the minimum value of `(2x)/(y^3)+(x^3y)/(3)+(4y^2)/(9x^4)` is ___________.

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The correct Answer is:
2
As x,y `in` R and `xy gt 0`, so x and y will be same sign.
Therefore, all the quantities `(2x)/(y^(3)), (x^(3) y)/(3), (4y^(2))/(9x^(4))` are positive
Now A.M `ge` G.M
`implies (2x)/(y^(3)) + (x^(3) y)/(3) + (4y^(2))/(9x^(4)) ge 3 (((2x)/(y^(3)))((x^(3) y)/(3))(4y^(2))/(9x^(4)))^(1//3)`
`= 3 xx (2)/(3) = 2`
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