Đáp án:
`a^4+b^4+2>=4ab(**)`
`<=>a^4-2a^2b^2+b^4+2+2a^2b^2>=4ab`
`<=>(a^2-b^2)^2+2a^2b^2-4ab+2>=0`
`<=>(a^2-b^2)^2+2(a^2b^2-2ab+1)>=0`
`<=>(a^2-b^2)^2+2(ab-1)^2>=0`(luôn đúng)
`=>(**)` được chứng minh.
Dấu "=" xảy ra khi:
$\begin{cases}a^2=b^2\\ab=1\end{cases}\\\Leftrightarrow\begin{cases}\left[ \begin{array}{l}a=b\\a=-b\end{array} \right.\\ab=1\end{cases}\\\Leftrightarrow\left[ \begin{array}{l}a.a=1\\a.(-a)=1\end{array} \right.\\\Leftrightarrow\left[ \begin{array}{l}a^2=1\\a^2=-1(l)\end{array} \right.\\\Leftrightarrow\left[ \begin{array}{l}a=b=1\\a=b=-1\end{array} \right.$
