Exam 1

  1. Question

    Compute the Hessian of the function f(x1,x2)=7x12+5x1x2+3x22 \begin{aligned} f(x_1, x_2) = 7 x_1^{2} + 5 x_1 x_2 + 3 x_2^{2} \end{aligned} at (x1,x2)=(1,4)(x_1, x_2) = (1, 4). What is the value of the upper left element?


    1. 66
    2. 77
    3. 1414
    4. 55
    5. −19-19

    Solution

    The first-order partial derivatives are f′1(x1,x2)=14x1+5x2f′2(x1,x2)=5x1+6x2 \begin{aligned} 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{aligned} and the second-order partial derivatives are f″11(x1,x2)=14f″12(x1,x2)=5f″21(x1,x2)=5f″22(x1,x2)=6 \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) &= 6 \end{aligned}

    Therefore the Hessian is f″(x1,x2)=(14556) \begin{aligned} f''(x_1, x_2) = \left( \begin{array}{rr} 14 & 5 \\ 5 & 6 \end{array} \right) \end{aligned} independent of x1x_1 and x2x_2. Thus, the upper left element is: f″11(1,4)=14f''_{11}(1, 4) = 14.


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