Question

cm \(x^4+y^4+z^2+1\ge2x\left(xy^2-x+z+1\right)\)

Answer 1

Lời giải:

Có: \(x^4+y^4+z^2+1\geq 2x(xy^2-x+z+1)\)

\(\Leftrightarrow x^4+y^4+z^2+1-2x^2y^2+2x^2-2xz-2x\geq 0\)

\(\Leftrightarrow (x^4+y^4-2x^2y^2)+(z^2+x^2-2xz)+(x^2+1-2x)\geq 0\)

\(\Leftrightarrow (x^2-y^2)^2+(z-x)^2+(x-1)^2\geq 0\)

Điều trên luôn đúng do \((x^2-y^2)^2\geq 0; (z-x)^2\geq 0; (x-1)^2\geq 0\)

Ta có đpcm

Dấu "=" xảy ra khi \(\left\{\begin{matrix} x^2-y^2=0\\ z-x=0\\ x-1=0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x=1\\ z=1\\ y=\pm 1\end{matrix}\right.\)