`P=x^4.y^4+x^4+y^4+1`
Ta có:`x^2+y^2``=(x+y)^2-2xy``=10-2xy`
`=>x^4+y^4=(x^2+y^2)^2-2x^2y^2=(10-2xy^2)-2(xy)^2=100-40xy+2(xy)^2`
`=>P=(xy)^4+2(xy)^2-40xy+101=[(xy)^4-8(xy)^2+16]+10[(xy)^2-4y+4]+45=[(xy^2-4]^2+10(xy-2)^2+45`
`=>P>=45`
Dấu `=` xảy ra khi `xy=2`
Mà ta có `x+y=\sqrt{10}`
`=>x=\sqrt{10}-y`
`=>xy=\sqrt{10}y-y^2=2`
`=>y^2-\sqrt{10}y+2=0`
`\Delta=10-8=2`
`=>y=\frac{\sqrt{10}+\sqrt{2}}{2}`
`=>x=\frac{4}{\sqrt{10}+\sqrt{2}}=\frac{\sqrt{10}-\sqrt{2}}{2}`
Vậy `P_{min}=45<=>x=\frac{\sqrt{10}-\sqrt{2}}{2};y=\frac{\sqrt{10}+\sqrt{2}}{2}`
