Please answer the following questions using Big-O notation

a) Show that if d(n) is O(f(n)) and e(n) is O(g(n)), then d(n) + e(n) is O(f(n) + g(n)).

b) 2ⁿ⁺¹ + n³ is O(2ⁿ) (Prove or disprove)

c) Prove that 2ⁿ is O(n!)

d) log(n!) is O(n log n) (Prove or disprove)

e) Show that if d(n) is O(f(n)) and e(n) is O(g(n)), then d(n) e(n) is not necessarily O(f(n) g(n)).