Question

The set of real values of $x$ satisfying
${\log }_{1/2}(x^2-6x+12)\ge -2$ is

• A. $(-\infty ,2]$
• B. $[2,4]$
• C. $[4,\infty )$
• D. none of these

Answer 1

The correct option is B $[2,4]$

As $x^2-6x+12>0$ for all $x$.
Therefore, ${\log }_{1/2}(x^2-6x+12)$ is defined for all $x\in R$
Now,
${\log }_{1/2}(x^2-6x+12)\ge -2$
$\Rightarrow x^2-6x+12\le {\left(\frac{1}{2}\right)}^{-2}$
$\Rightarrow x^2-6x+8\le 0\Rightarrow (x-2)(x-4)\le 0\Rightarrow x\in [2,4]$