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Applications of Derivative — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivative3 Marks
Question
Find the interval for which function $f(x)=2 \log (x-2)-x^2+4 x+1$ is increasing.
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Answer

Here $f(x)=2 \log (x-2)-x^2+4 x+1$
$\Rightarrow y=2 \log (x-2)-x^2+4 x+1$
$\Rightarrow \frac{d y}{d x} =2 \frac{2}{(x-2)}-2 x+4$
$ =2\left[\frac{1}{x-2}-x+2\right]=2\left[\frac{1}{(x-2)}-(x-2)\right]$
$\Rightarrow \frac{d y}{d x} =2\left[\frac{1-(x-2)^2}{(x-2)}\right]=\frac{2(x-1)(3-x)}{(x-2)}$
because given function $f(x)$ is increasing
So $, \frac{d y}{d x} > 0$
$\Rightarrow \frac{2(x-1)(3-x)}{(x-2)} > 0$
$\Rightarrow 2(x-1)(3-x) > 0$
$\Rightarrow (x-1)(3-x) > 0\ [\because 2>0]$
$\Rightarrow 1< x < 3$
Hence interval is $1 < x < 3$.

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