Problem 11.3 (Maximizing the area of a rectangle inscribed in a circle). We want to find the largest rectangle, with sides of length = and y, that can be inscribed inside a circle of diameter 1. In other words, we want to maximize the function f(z,y) = xy, subject to the condition that \\/a2+y2 = 1. (a) Write the function f as a quadratic form @ : R> R, i.e. as a function of the type f(r1,22) = Q(x) = x" Ax (with z; = x and 3 = y), where A is a (2 x 2) real and symmetric matrix. (b) What are the eigenvalues A\\; and A2 of A? Also, find an orthonormal basis B = {vi,v2} of R? that is made of eigenvectors of A. Rewrite the quadratic form @Q(x) as a function of the coordinates ; and co of the generic vector x R? with respect to the basis 3, i.e. as a function of [x] 5= [c1 ]T. How does the constraint y/2?+z3 = 1 translate in terms of ; and a? Also, what is the shape of the surface z = Q(x)? (c) Finally, find the values x and y that maximize the the area of the rectangle insctibed in the circle of diameter 1. For these values of = and , what is the area of the rectangle