Exam 1

  1. Question

    Compute the Hessian of the function
    f( x1 , x2 )=7 x1 2 +5 x1 x2 +3 x2 2

    at ( x1 , x2 )=(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
    f '1 ( x1 , x2 ) = 14 x1 +5 x2 f '2 ( x1 , x2 ) = 5 x1 +6 x2

    and the second-order partial derivatives are
    f"11 ( x1 , x2 ) = 14 f"12 ( x1 , x2 ) = 5 f"21 ( x1 , x2 ) = 5 f"22 ( x1 , x2 ) = 6

    Therefore the Hessian is
    f"( x1 , x2 )=( 145 56 )

    independent of x1 and x2 . Thus, the upper left element is: f"11 (1,4)=14.

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