Question

If $x^2-4x+{\log }_{\frac{1}{2}}a$ does not have two distinct roots, then the maximum value of $a$

• A. $-\frac{1}{4}$
• B. $\frac{1}{4}$
• C. $\frac{1}{16}$
• D. none

Answer 1

The correct option is C $\frac{1}{16}$

Applying the given condition, we get
$b^2-4ac\le 0$
$16-4lo{g}_{1/2}\left(a\right)\le 0$
$lo{g}_{1/2}\left(a\right)\ge 4$
$\therefore a\le {\left(\frac{1}{2}\right)}^4$ and $a>0$
Hence,
$a\in (0,\frac{1}{16}]$
Hence, maximum value of $a$ is $\frac{1}{16}$.