Question

Chứng minh : \(\left(x-1\right)\left(x-3\right)\left(x-4\right)\left(x-6\right)+9\ge0\)

Answer 1

_Solution

We have:

\(\left(x-1\right)\left(x-3\right)\left(x-4\right)\left(x-6\right)+9\ge0\)

\(\Leftrightarrow\left(x-1\right)\left(x-6\right).\left(x-3\right)\left(x-4\right)+9\ge0\)

\(\Leftrightarrow\left(x^2-7x+6\right).\left(x^2-7x+12\right)+9\ge0\)\(\left(1\right)\)

Replace: \(x^2-7x+6=t\)

\(\left(1\right)\Leftrightarrow t.\left(t+6\right)+9\ge0\)

\(\Leftrightarrow t^2+6t+9\ge0\)

\(\Leftrightarrow\left(t+3\right)^2\ge0\)

=> We finish proving this Inequality.