Exam 1

1. Question

   Compute the Hessian of the function
   

   ${\displaystyle \begin{array}{c}\multicolumn{1}{c}{f(x_1,x_2)=7x_1^2+5x_1{x}_2+3x_2^2}\end{array}}$

   
   at $(x_1,x_2)=(1,4)$. 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
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{f{\text{'}}_1(x_1,x_2)}& =\hfill & 14x_1+5x_2\hfill \\ \multicolumn{1}{c}{f{\text{'}}_2(x_1,x_2)}& =\hfill & 5x_1+6x_2\hfill \end{array}}$

   
   and the second-order partial derivatives are
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{f"{}_{11}(x_1,x_2)}& =\hfill & 14\hfill \\ \multicolumn{1}{c}{f"{}_{12}(x_1,x_2)}& =\hfill & 5\hfill \\ \multicolumn{1}{c}{f"{}_{21}(x_1,x_2)}& =\hfill & 5\hfill \\ \multicolumn{1}{c}{f"{}_{22}(x_1,x_2)}& =\hfill & 6\hfill \end{array}}$

   
   
   Therefore the Hessian is
   

   ${\displaystyle \begin{array}{c}\multicolumn{1}{c}{f"(x_1,x_2)=\left(\begin{array}{cc}\hfill 14& \hfill 5\\ \hfill 5& \hfill 6\end{array}\right)}\end{array}}$

   
   independent of $x_1$ and $x_2$. Thus, the upper left element is: $f"{}_{11}(1,4)=14$.
   
   
   1. False
   2. False
   3. True
   4. False
   5. False