2. (15 points) Select all of the correct answer(s) in each of the following problems. (a) (3 points) Suppose that f: D → R is a continuous function defined on a

closed and bounded set D. Which (if any) of the following must also be true? • f has a global extrema, and it occurs either at a critical point or a boundary point. • f has a critical point. • f must have either a global maximum or a global minimum, but possibly not both. • f cannot have a saddle point. (b) (3 points) Let u and v be two non-zero vectors in R", and assume that u = 0. Which (if any) of the following must also necessarily be true. • The span of u and v must have dimension 0. • The projection of u onto v is the zero vector. • The vectors u and v are parallel. • the vectors u and v are linearly independent. (c) (3 points) Which (if any) of the following are critical points of the function f(x, y) = 2 + y - 6xy: • (0,0) • (-1,1) (1,1) • (2,1)/n(d) (3 points) Which (if any) of the following matrices are positive definite? • 0 22 22 • (-¹) . 2 2 (e) (3 points) Let f(r) be a function whose every derivative exists and is continuous. If the third degree Taylor polynomial of f centered at zero is given by 2x + 3x³, which of the following (if any) must also be true. • The third degree Taylor polynomial centered at 1 is 2(x-1) + 3(x - 1)³ • f'(r) has a critical point at x = 0. . f(r) has an inflection point at x = 0. f(r) is increasing near z = 0.