Question

The domain of the definition of the function $\sqrt{{\log }_{10}\left(\frac{5x-x^2}{4}\right)}$ is

• A. $[1,4]$
• B. $(1,4]$
• C. $(0,5)$
• D. $[0,5]$

Answer 1

The correct option is A $[1,4]$

For function to be defined,
${\log }_{10}\left(\frac{5x-x^2}{4}\right)\ge 0\text{and}\frac{5x-x^2}{4}>0\Rightarrow \frac{5x-x^2}{4}\ge 1\text{and}5x-x^2>0$
$\Rightarrow 5x-x^2-4\ge 0\text{and}{x}^2-5x<0\Rightarrow x^2-5x+4\le 0\text{and}x(x-5)<0\Rightarrow (x-4)(x-1)\le 0\text{and}0<x<5\Rightarrow 1\le x\le 4\text{and}0<x<5\Rightarrow x\in [1,4]$