## Exam 1

1. #### Question

Compute the Hessian of the function
 $\begin{array}{c}\multicolumn{1}{c}{f\left({x}_{1},{x}_{2}\right)=7{x}_{1}^{2}+5{x}_{1}{x}_{2}+3{x}_{2}^{2}}\end{array}$

at $\left({x}_{1},{x}_{2}\right)=\left(1,4\right)$. What is the value of the upper left element?

1. $6$
2. $7$
3. $14$
4. $5$
5. $-19$

#### Solution

The first-order partial derivatives are
 $\begin{array}{ccc}\multicolumn{1}{c}{f{\text{'}}_{1}\left({x}_{1},{x}_{2}\right)}& =\hfill & 14{x}_{1}+5{x}_{2}\hfill \\ \multicolumn{1}{c}{f{\text{'}}_{2}\left({x}_{1},{x}_{2}\right)}& =\hfill & 5{x}_{1}+6{x}_{2}\hfill \end{array}$

and the second-order partial derivatives are
 $\begin{array}{ccc}\multicolumn{1}{c}{f"{}_{11}\left({x}_{1},{x}_{2}\right)}& =\hfill & 14\hfill \\ \multicolumn{1}{c}{f"{}_{12}\left({x}_{1},{x}_{2}\right)}& =\hfill & 5\hfill \\ \multicolumn{1}{c}{f"{}_{21}\left({x}_{1},{x}_{2}\right)}& =\hfill & 5\hfill \\ \multicolumn{1}{c}{f"{}_{22}\left({x}_{1},{x}_{2}\right)}& =\hfill & 6\hfill \end{array}$

Therefore the Hessian is
 $\begin{array}{c}\multicolumn{1}{c}{f"\left({x}_{1},{x}_{2}\right)=\left(\begin{array}{cc}\hfill 14& \hfill 5\\ \hfill 5& \hfill 6\end{array}\right)}\end{array}$

independent of ${x}_{1}$ and ${x}_{2}$. Thus, the upper left element is: $f"{}_{11}\left(1,4\right)=14$.

1. False
2. False
3. True
4. False
5. False