Blaming your teacher isn't going to help you at all. No matter how how bad your teacher is, you surely should be able to look up thedefinitionof "linearly independent". For functions it is that the only way you can have , for all x, is to have .

a) If , what are f'(x), f"(x), and f'''(x)? Suppose for all x. Taking four different values for x will give you four linear equations to solve for , , , and .

b) If , what are and . Suppose for all x. Taking three different values for x will give you three linear equations to solve.

c) I assume that by " you mean , not .

Again, suppose you have . Taking three different values for x will give you three linear equations to solve.