Question

[Nguyễn Hoài Phương]

1. cho \(x,y\ge1\) c/m \(x\sqrt{y-1}+y\sqrt{x-1}\le xy\)
2. cho \(a,b,c\in R\) ,\(\hept{\begin{cases}a+b+c=2\\ab+bc+ac=1\end{cases}}\)C/m \(0\le a,b,c\le\frac{4}{3}\)

Answer 1

[HeroZombie]

1)Áp dụng BĐT Cô si ta có:

\(x\sqrt{y-1}\le\frac{x\left(y-1+1\right)}{2}=\frac{xy}{2}\)

\(y\sqrt{x-1}\le\frac{y\left(x-1+1\right)}{2}=\frac{xy}{2}\)

Cộng thei vế 2 BĐT cùng chiều ta có:

\(VT\le\frac{xy}{2}+\frac{xy}{2}=\frac{2xy}{2}=xy=VP\)

Khi x=y

Answer 2

[HeroZombie]

Ta có BĐT \(\left(a+b+c\right)^2\ge3\left(ab+bc+ca\right)\)

\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\) (đúng)

\(\Rightarrow2^2\ge3\cdot1\Rightarrow\frac{4}{3}\ge a,b,c\ge0\)

Khi a=b=c