Đáp án:
a,
`x^2 + 2xy + y^2 = 16`
`=> x^2 + y^2 = 16 - 2xy`
`(x^2 + 3y)(y^2 + 3x) + 21xy`
`= x^2y^2 + 3y^3 + 3x^3 + 9xy + 21xy`
`= x^2y^2 + 3y^3 + 3x^3 + 30xy`
`= x^2y^2 + 3(x + y)(x^2 - xy + y^2) + 30xy`
`= x^2y^2 + 12(x^2 - xy + y^2) + 30xy`
`= x^2y^2 + 12(16 - 2xy - xy) + 30xy`
`= x^2y^2 + 192 - 36xy + 30xy`
`= x^2y^2 - 6xy + 192`
`= (xy)^2 - 2.xy . 3 + 9 + 183`
`= (xy - 3)^2 + 183 ≥ 183`
Dấu "=" xảy ra `<=> xy - 3 = 0 <=> xy = 3 <=> (x,y) = (1,3);(3,1)`
Vậy Min P là `183 <=> (x,y) = (1,3) ; (3,1)`
b, Ta có :
`2\sqrt{xy} ≤ x + y`
`=> 2\sqrt{xy} ≤ 4`
`=> \sqrt{xy} ≤ 2`
`=> 0 ≤ xy ≤ 4`
`=> (xy - 3)^2 + 183 ≤ (0 - 3)^2 + 183 = 192`
Dấu "=" xảy ra `<=> (x,y) = (0,4) ; (4,0)`
Vậy MaxP là `192 <=> (x,y) = (0,4) ; (4,0)`
