# Mr. Isulo, a famous alien computer scientist, has designed efficient

algorithms to solve two problems whose solutions will help the Martian government preserve the color of their home planet: • One algorithm performs n^4 + 3n^2 + 5 steps, where n is the size of the algorithms' input. • The other algorithm performs 2n^2 + 10 steps, where n is the size of the algorithms' input.

Help Mr. Isulo convince the Martian government to pay him for his work by formally proving the following two theorems.

Recall that f ∈ O(g) if ∃n0 ∈ N ∃c ∈ R+ ∀n ∈ N, n ≥ n0 →f(n) ≤ cg(n).

a. [8 marks] n^4 + 3n^2 + 5 ∈ O(n^4 ).

b. [8 marks] 2n^2 + 10 ∈ O(3n^2 + 7n). Hint: first choose a value of c that you think will work, and once you have c do some algebra to determine how to choose n0.

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