Prove the following bounds:

(a) f(x) = a_nxⁿ + a_n1x ⁿ¹ + + a₁x + a₀ is O(x ⁿ ).

(b) 2 + 4 + + 2(n 1) + 2n is (n²).

(c) n! = 1 2 3 (n 1) n is O(nⁿ).

(d) log(n!) is O(n log(n)).