In parts (a)(e) determine whether the statement is true or

Chapter 7, Problem T2

(choose chapter or problem)

In parts (a)(e) determine whether the statement is true or false, and justify your answer. (a) A quadratic form must have either a maximum or minimum value. (b) The maximum value of a quadratic form xT Ax subject to the constraint x = 1 occurs at a unit eigenvector corresponding to the largest eigenvalue of A. (c) The Hessian matrix of a function f with continuous secondorder partial derivatives is a symmetric matrix. (d) If (x0, y0) is a critical point of a function f and the Hessian of f at (x0, y0) is 0, then f has neither a relative maximum nor a relative minimum at (x0, y0). (e) If A is a symmetric matrix and det(A) < 0, then the minimum of xT Ax subject to the constraint x = 1 is negative Suppose that the temperature at a point (x, y) on a metal plate is T (x, y) = 4x2 4xy + y2. An ant walking on the plate traverses a circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant?