a)Áp dụng BĐT Cauchy-Schwarz ta có:
\(VT=\dfrac{1}{x^3\left(y+z\right)}+\dfrac{1}{y^3\left(x+z\right)}+\dfrac{1}{z^3\left(x+y\right)}\)
\(=\dfrac{x^2y^2z^2}{x^3\left(y+z\right)}+\dfrac{x^2y^2z^2}{y^3\left(x+z\right)}+\dfrac{x^2y^2z^2}{z^3\left(x+y\right)}\)
\(=\dfrac{y^2z^2}{x\left(y+z\right)}+\dfrac{x^2z^2}{y\left(x+z\right)}+\dfrac{x^2y^2}{z\left(x+y\right)}\)
\(\ge\dfrac{\left(xy+yz+xz\right)^2}{2\left(xy+yz+xz\right)}=\dfrac{xy+yz+xz}{2}\ge\dfrac{3\sqrt[3]{\left(xyz\right)^2}}{2}=\dfrac{3}{2}=VP\)
Xảy ra khi \(x=y=z=1\)
b)Áp dụng BĐT Cauchy-Schwarz ta có:
\(y^2=\left(\sqrt{x-2}+\sqrt{4-x}\right)^2\)
\(\le\left(1+1\right)\left(x-2+4-x\right)=4\)
\(\Rightarrow y^2\le4\Rightarrow y\le2\)
Khi \(x=3\)