$\therefore g\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{k\sqrt {k + 1} \,\,\,,\,\,\,\,0 \le x \le 3}\\
{mx + 2\,\,\,\,\,,\,\,\,\,3 < x \le 5}
\end{array}} \right.$
At $x = 3,\,\,\,\,\,\,\,\,\,RHL = 3m + 2$
and at $x = 3,LHL = 2k$
$\therefore \,\,\,\,\,2k = 3m + 2\,\,\,\,.....\left( i \right)$
Also, $g'\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{\frac{k}{{2\sqrt {x + 1} }}\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,0 \le x < 3}\\
{m\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,3 < x \le 5\,}
\end{array}} \right.$
$\therefore L\left\{ {g'\left( 3 \right)} \right\} = \frac{k}{4}$ and $R\left\{ {g'\left( 3 \right)} \right\} = m$
$ \Rightarrow $ $\frac{k}{4} = m$ i.e. $k=4m$ ........$(ii)$
On solving Eqs. $(i)$ and $(ii)$, we get
$k = \frac{8}{5},m = \frac{2}{5}$
$ \Rightarrow $ $k + m = 2$
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