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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 $f(x)=2 \log (x-2)-x^2+4 x+1$
$\Rightarrow \quad y=2 \log (x-2)-x^2+4 x+1$
$
\begin{array}{l}

\begin{aligned}
\Rightarrow \quad \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 \quad \frac{d y}{d x} & =2\left[\frac{1-(x-2)^2}{(x-2)}\right]=\frac{2(x-1)(3-x)}{(x-2)}
\end{aligned}
\end{array}
$because given function $f(x)$ is increasing
\[\begin{array}{l}\text { So, } \quad \frac{d y}{d x} > 0 \\\Rightarrow  \frac{2(x-1)(3-x)}{(x-2)} > 0 \\\Rightarrow  2(x-1)(3-x) > 0 \\\Rightarrow (x-1)(3-x) > 0[\because 2>0] \\\Rightarrow  1<  x < 3\end{array}\]
Hence interval is $1 < x < 3$.

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