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

Chapter 7, Problem T1

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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 . Find the maximum and minimum values of the following quadratic form subject to the stated constraint, and specify the points at which those values are attained. w = 2x2 + y2 + z2 + 2xy + 2xz; x2 + y2 + z2 = 1