Question

Giải:

\(\sqrt{\frac{x}{2}-\frac{22}{21}}-\sqrt[3]{x^3-3^2+\frac{23}{27}}=1\) 

Answer 1

-Xét \(x\ge y\ge z\). Dễ cm bđt đúng

-Xét \(x\ge z\ge y\)

Đặt x=z+a, z=y+b với \(a,b\ge0\)

=>x=y+a+b

BĐT\(< =>\frac{x-y}{y\left(y+1\right)}\ge\frac{x-z}{x\left(x+1\right)}+\frac{z-x}{z\left(z+1\right)}\)

<=>\(\frac{a+b}{y\left(y+1\right)}\ge\frac{a}{x\left(x+1\right)}+\frac{b}{z\left(z+1\right)}\)

Vì \(x\ge z\ge y=>x\left(x+1\right)\ge z\left(z+1\right)\ge y\left(y+1\right)\)

\(=>\frac{a}{y\left(y+1\right)}\ge\frac{a}{x\left(x+1\right)},\frac{b}{y\left(y+1\right)}\ge\frac{b}{z\left(z+1\right)}\)

=>\(\frac{a+b}{y\left(y+1\right)}\ge\frac{a}{x\left(x+1\right)}+\frac{b}{z\left(z+1\right)}\)=>bđt cần cm đúng=>đpcm