Question

The domain of $f\left(x\right)=\sqrt{\log \left(\frac{5x-x^2}{6}\right)}$ is

• A. $2\le x\le 3$
• B. $3\le x\le 4$
• C. $-3\le x\le -2$
• D. $1\le x\le 3$

Answer 1

The correct option is A $2\le x\le 3$

The function is defined for all values of x for which $\frac{5x-x^2}{6}>0$ ...(1)

and $\log \left[\frac{(5x-x^2)}{6}\right]\ge 0=\log 1$ ...(2)

(1)$\Rightarrow x(5-x)>0$ or $x(x-5)<0$ ...(3)

(2)$\Rightarrow \frac{(5x-x^2)}{6}\ge 1i.e.x^2-5x+6\le 0$ ...(4) or

$x(x-5)<0$ and $(x-2)(x-3)\le 0$

$\therefore D_1=[0,5],D_2=[2,3]$

Hence the inequalities (1)

and (2) will hold for $D_1\cap D_{2}i.e.$ if
$2\le x\le 3$ or $[2,3]$

(Draw $D_1$ and $D_2$ on real line and take the common part of both.)