Exam 1

  1. Question

    Compute the Hessian of the function
    f( x1 , x2 )=-7 x1 2 -5 x1 x2 -4 x2 2

    at ( x1 , x2 )=(-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
    f '1 ( x1 , x2 ) = -14 x1 -5 x2 f '2 ( x1 , x2 ) = -5 x1 -8 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 ) = -8

    Therefore the Hessian is
    f"( x1 , x2 )=( -14-5 -5-8 )

    independent of x1 and x2 . Thus, the upper left element is: f"11 (-2,2)=-14.

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