Question

If the range of values of ${}^{\prime }{a}^{\prime }$ for which $f(x)={\left(log\right)}_a\left(4ax-x^2\right)$ is strictly increasing $\forall x\in \left[\frac{3}{2},2\right]$ is $(p,q]\cup (r,\infty )$ then the value of $(2p+4q+r)$ equals

Answer 1

Answer: 5

$f^{{}^{\prime }}(x)=\frac{(4a-2x)}{\left(4ax-x^2\right)loga}\ge 0$
$f^{{}^{\prime }}(x)=\frac{(x-2a)}{x(x-4a)loga}\ge 0$
$C-1:a\in (1,\infty )\Rightarrow \frac{(x-2a)}{x(x-4a)}\ge 0$

$x\in (0,2a)$ so always $\uparrow$
$C-2:a\in (0,1)$
$\frac{(x-2a)}{x(x-4a)}<0$

$x\in (2a,4a)$
$2a\le \frac{3}{2}\Rightarrow a\le \frac{3}{4}$ and $4a>2\Rightarrow a>\frac{1}{2}$
So $a\in (\frac{1}{2},\frac{3}{4}]\cup [1,\infty )$