Áp dụng BĐT AM-GM ta có:
\(\frac{x^4}{y+3z}+\frac{y+3z}{16}+\frac{1}{4}+\frac{1}{4}\ge4\sqrt[4]{\frac{x^4}{y+3z}\cdot\frac{y+3z}{16}\cdot\frac{1}{4}\cdot\frac{1}{4}}=x\)
\(\Rightarrow\frac{x^4}{y+3z}\ge x-\frac{y+3z}{16}-\frac{1}{2}\).Tương tự ta có:
\(\frac{y^4}{z+3x}\ge y-\frac{z+3x}{16}-\frac{1}{2};\frac{z^4}{x+3y}\ge z-\frac{x+3y}{16}-\frac{1}{2}\)
Cộng theo vế ta có:
\(P\ge\frac{3}{4}\left(x+y+z\right)-\frac{3}{2}\ge\frac{3}{4}\cdot3-\frac{3}{2}=\frac{3}{4}\)
Dấu "=" khi x=y=z=1
xin cho mình hỏi sao x+y+z lại\(\ge\)xy+yz+zx vậy
Áp dụng bất đẳng thức AM-GM, ta có: \(a^2+b^2+c^2\ge ab+bc+ca\)
<=>\(a^2+b^2+c^2+2ab+2bc+2ca\ge3\left(ab+bc+ca\right)\)
<=>\(\left(a+b+c\right)^2\ge9\)
<=>\(a+b+c\ge3\)