Question

We call a a good number if inequality $\frac{{2x}^2+2x+3}{x^2+x+1}\le a$ is satisfied for any real x. Find the smallest good integral number

Answer 1

$\frac{{2x}^2+2x+3}{x^2+x+1}\le a$
$\Rightarrow x^2(2-a)+x(2-a)+3-a\le 0$
$\Rightarrow D\ge 0and2-a<0$
$\Rightarrow (2-a{)}^2-4(2-a\left)\right(3-a)\ge 0$
$\Rightarrow (2-a)(3a-10)\ge 0$
$\Rightarrow 3a-10\ge 0$ ($\because 2-a<0$)
$\Rightarrow a\in [\frac{10}{3},\infty )$
So, the smallest good integer is 4