Question

The function $f\left(x\right)=2\ln (x-2)-x^2+4x+1$ increases in the interval-

• A. $(1,2)$
• B. $(2,3)$
• C. $(-\infty ,-1)$
• D. $(2,4)$

Answer 1

The correct option is B $(2,3)$

$f^{\prime }\left(x\right)>0$ for monotonically increasing function.
Hence
$\frac{2}{x-2}-2x+4>0$
Or
$\frac{2}{x-2}>2x-4$
Or
$\frac{1}{x-2}>x-2$
Or
$(x-2{)}^2<1$
$-1<x-2<1$
Or
$1<x<3$
Hence
$xϵ(1,3)$
However $x-2>0$ since input of logarithmic function cannot be negative.
$(x-2)>0$ or
$x>2$
Hence
$xϵ(2,3)$.