## Exam 1

1. #### Question

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

1. $-8$
2. $-5$
3. $-16$
4. $-14$
5. $-4$

#### Solution

The first-order partial derivatives are \begin{aligned} f'_1(x_1, x_2) &= -14 x_1 -5 x_2 \\ f'_2(x_1, x_2) &= -5 x_1 -8 x_2 \end{aligned} and the second-order partial derivatives are \begin{aligned} 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) &= -8 \end{aligned}

Therefore the Hessian is \begin{aligned} f''(x_1, x_2) = \left( \begin{array}{rr} -14 & -5 \\ -5 & -8 \end{array} \right) \end{aligned} independent of $x_1$ and $x_2$. Thus, the upper left element is: $f''_{11}(-2, 2) = -14$.

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