help me

Question

Cho

$x;y;z\in R$ thỏa mãn

$8^{x}+8^{y}+8^z=3$

CMR

$\frac{4^x}{3-4^{x}}+\frac{4^y}{3-4^y}+\frac{4^z}{3-4^z}\geq \frac{3}{2}$

Ruanyu Jian · Answer 1 · 6/29/2016 11:39:39 AM

Đặt

$a=2^x,b=2^y,c=2^z\Rightarrow a,b,c>0 và a^3+b^3+c^3=3$

BĐT cần c/m tương đương với

$\frac{a^2}{3-a^2}+\frac{b^2}{3-b^2}+\frac{c^2}{3-c^2}\geq \frac{3}{2}\Leftrightarrow \frac{a^3}{a(3-a^2)}+\frac{b^3}{b(3-b^2)}+\frac{c^3}{c(3-c^2)}\geq \frac{3}{2}$

Theo bđt Cô-si ta có

$2\left[ {a(3-a^2)} \right]^2=2a^2(3-a^2)(3-a^2)\leq (\frac{2a^2+3-a^2+3-a^2}{3})^3=8\Rightarrow a(3-a^2)\leq 2\Rightarrow \frac{a^3}{a(3-a^2)}\geq \frac{a^3}{2}$

Tương tự ta có