2. The function $g$ is defined for $x > 0$ with $g ( 1 ) = 2 , g ^ { \prime } ( x ) = \sin \left( x + \frac { 1 } { x } \right)$, and $g ^ { \prime \prime } ( x ) = \left( 1 - \frac { 1 } { x ^ { 2 } } \right) \cos \left( x + \frac { 1 } { x } \right)$.\\
(a) Find all values of $x$ in the interval $0.12 \leq x \leq 1$ at which the graph of $g$ has a horizontal tangent line.\\
(b) On what subintervals of $( 0.12,1 )$, if any, is the graph of $g$ concave down? Justify your answer.\\
(c) Write an equation for the line tangent to the graph of $g$ at $x = 0.3$.\\
(d) Does the line tangent to the graph of $g$ at $x = 0.3$ lie above or below the graph of $g$ for $0.3 < x < 1$ ? Why?

\begin{center}
\begin{tabular}{ | c | | c | c | c | c | c | c | c | }
\hline
$t$ & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\
\hline
$P ( t )$ & 0 & 46 & 53 & 57 & 60 & 62 & 63 \\
\hline
\end{tabular}
\end{center}

\includegraphics[max width=\textwidth, alt={}, center]{d04a7a11-0c12-4aac-bb2f-8c47963b3289-3_278_850_807_949}\\