If `x ,y in R` and `x^2+y^2+x y=1,` then find the minimum value of `x^3y+x y^3+4.`

Explanation

`x^(2)+y^(2)+xy=1`
Let `x=r cos theta, y=r sin theta`
`therefore r^(2)cos^(2)theta+r^(2)sin^(2)theta+r^(2)(cos theta)(sin theta)=1`
`therefore r^(2)sin theta cos theta=1-r^(2)`
`therefore r^(2)=(2)/(2+sin 2theta)`
Now, `-1 lesin 2 theta le1`
`therefore 1 le2+sin 2thetale3`
`therefore (1)/(3)lt(1)/(2+sin 2theta)le1`
`therefore(2)/(3)le(2)/(2+sin 2theta)le2`
or `(2)/(3)ler^(2)le2`
Now `E=x^(3)y+xy^(3)+4`
`=xy(x^(2)+y^(2))+4`
`=r^(2)sintheta cos theta(r^(2)cos^(2)theta+r^(2)sin^(2)theta)+4`
`=r^(2)(1-r^(2))+4`
`=(!7)/(4)-(r^(2)-(1)/(2))^(2)`
`therefore E_("min")=(17)/(4)-(2-(1)/(2))^(2)=2`