Question

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

• A. (1, 2)
• B. (2, 3)
• C. (2, 4)
• D. $(0,\infty )$

Answer 1

The correct option is B (2, 3)

$f\left(x\right)=2\log (x-2)-x^2+4x+1$
Clearly f(x) is defined for $x>2$
$f^{\prime }\left(x\right)=\frac{2}{x-2}-2x+4$
$\Rightarrow f^{\prime }\left(x\right)=-2\frac{(x-1)(x-3)}{x-2}$
For f(x) to be increasing, f'(x)>0
$\Rightarrow -2\frac{(x-1)(x-3)}{x-2}>0$
$\Rightarrow (x-1)(x-3)<0$
But f(x) is defined for x>2
Hence, $x\in (2,3)$