Let f(x) = x1^2 + 7x^2 2 + 7x3^2 - 8x1x2 - 4x2x3 - 8x1x3: Hint: The eigenvalues of the matrix A

of the quadratic form are -3 and 9.

a. Compute the max of f(x) subject to x1^2 + x2^ 2 + x3^2 = 1. Write one unit vector x at which

the max is obtained.

b. Compute the min of f(x) subject to x1^2 + x2^ 2 + x3^2 = 1. Write one unit vector x at which

the min is obtained

Let f(x1; x2; x3) = 3x1^2 + 5x2^ 2 - 2x3^2.

a. Find max f(x1; x2; x3) subject to x1^2 + x2^ 2 + x3^2 = 1. What are the values of x1; x2; and x3

at which f is maximized? Justify your answer.

b. Find min f(x1; x2; x3) subject to x1^2 + x2^ 2 + x3^2 = 1. What are the values of x1; x2; and x3

at which f is minimized? Justify your answer.

c. Write the matrix A such that f(x) = xTAx. Write the eigenvalues of A. For each

eigenvalue, write corresponding unit eigenvector.

d. How do the eigenvalues and eigenvectors of A relate to the max/min of xTAx found in

(a-b)?