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

Chapter 7, Problem T3

(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 The accompanying figure shows the intersection of the surface z = x2 + 4y2 (called an elliptic paraboloid) and the surface x2 + y2 = 1 (called a right circular cylinder). Find the highest and lowest points on the curve of intersection.