
\begin{question}
Compute the Hessian of the function
\begin{eqnarray*}
  f(x_1, x_2) = 7 x_1^{2} + 5  x_1  x_2 + 3  x_2^{2}
\end{eqnarray*}
at $(x_1, x_2) = (1, 4)$.
What is the value of the upper left element?

\begin{answerlist}
  \item $6$
  \item $7$
  \item $14$
  \item $5$
  \item $-19$
\end{answerlist}\end{question}

\begin{solution}
The first-order partial derivatives are 
\begin{eqnarray*}
  f'_1(x_1, x_2) &=& 14 x_1 + 5 x_2  \\
  f'_2(x_1, x_2) &=& 5 x_1 + 6 x_2
\end{eqnarray*}
and the second-order partial derivatives are
\begin{eqnarray*}
  f''_{11}(x_1, x_2) &=& 14\\
  f''_{12}(x_1, x_2) &=& 5\\
  f''_{21}(x_1, x_2) &=& 5\\
  f''_{22}(x_1, x_2) &=& 6
\end{eqnarray*}

Therefore the Hessian is
\begin{eqnarray*}
  f''(x_1, x_2) = \left( \begin{array}{rr} 14 &  5 \\  5 &  6 \end{array} \right)
\end{eqnarray*}
independent of $x_1$ and $x_2$. Thus, the upper left element is:
$f''_{11}(1, 4) = 14$.


\begin{answerlist}
  \item False
  \item False
  \item True
  \item False
  \item False
\end{answerlist}
\end{solution}

%% \extype{schoice}
%% \exsolution{00100}
%% \exname{Hessian}
