Giup minh voi a minh đang can gap

`A=(a+b)/(\sqrt{a(3a+b}+\sqrt{b(3b+a)})`

`A=(4a+4b)/(2\sqrt{4a(3a+b}+2\sqrt{4b(3b+a)})`

Áp dụng BDT AM-GM

`2\sqrt{4a(3a+b)} <= 4a + (3a+b)`

`2\sqrt{4b(3b+a)} <= 4b + (3b+a)`

`⇒ A >= (4a+4b)/(4a+3a+b+4b+3b+a)`

`⇔ A >= (4a+4b)/(8a + 8b)`

`⇔ A >= 1/2`

Dấu "=" xảy ra khi `a=b`

Vậy, `Amin=1/2`

`color{purple}{@SHIN}`