Question

If the domain of the function
$f\left(x\right) =\left[log_{10} \left(\frac{5x - x^{2}}{4}\right)\right]^{-1/2}$ is $a \le x \le b$, then $a. b$ is

Answer 1

We have, $f\left(x\right) =\left[log_{10} \left(\frac{5x - x^{2}}{4}\right)\right]^{-1/2} ....\left(i\right)$
From (i), clearly f(x) is defined for those values of $x$ for which
$log_{10}\left[\frac{5x - x^{2}}{4}\right]\ge0$
$\Rightarrow \left(\frac{5x -x^{2}}{4}\right)\ge10^{0}$
$\Rightarrow \left(\frac{5x -x^{2}}{4}\right)\ge1$
$\Rightarrow x^{2} -5x +4 \le0$
$\Rightarrow \left(x -1\right)\left(x-4\right)\le0$
Hence, domain of the function is $\left[1,4\right]i e.1\le x\le4$