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Mark the correct alternative in the following:

Question:

Mark the correct alternative in the following:

The function $f(x)=2 \log (x-2)-x^{2}+4 x+1$ increases on the interval.

A. $(1,2)$

B. $(2,3)$

C. $((1,3)$

D. $(2,4)$

Solution:

Formula:- The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)>0$ for all $x \in(a, b)$

Given:-

$f(x)=2 \log (x-2)-x^{2}+4 x+1$

$\mathrm{d}\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{dx}}\right)=\frac{2}{\mathrm{x}-2}-2 \mathrm{x}+4=\mathrm{f}^{\prime}(\mathrm{x})$

$\Rightarrow f^{\prime}(x)=-\frac{2(x-1)(x-3)}{x-2}$

now for increasing

$f^{\prime}(x)>0$

$\Rightarrow-\frac{2(x-1)(x-3)}{x-2}<0$

$x-3<0$ and $x-2>0$

$x<3$ and $x>2$

$x \in(2,3)$

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