Exam 1

  1. Question

    Compute the Hessian of the function f(x1,x2)=7x125x1x24x22 \begin{aligned} f(x_1, x_2) = -7 x_1^{2} -5 x_1 x_2 -4 x_2^{2} \end{aligned} at (x1,x2)=(2,2)(x_1, x_2) = (-2, 2). What is the value of the upper left element?


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

    Solution

    The first-order partial derivatives are f1(x1,x2)=14x15x2f2(x1,x2)=5x18x2 \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 f11(x1,x2)=14f12(x1,x2)=5f21(x1,x2)=5f22(x1,x2)=8 \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 f(x1,x2)=(14558) \begin{aligned} f''(x_1, x_2) = \left( \begin{array}{rr} -14 & -5 \\ -5 & -8 \end{array} \right) \end{aligned} independent of x1x_1 and x2x_2. Thus, the upper left element is: f11(2,2)=14f''_{11}(-2, 2) = -14.


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