Question

Given the following function: f(n)=5n 12+4n 9 logn+36n 4 +20n 3 +50n+50. i) Prove or disprove that the above function is Ω(n 9 logn). Full and detailed step-by-step proof/disproof is required. ii) Is the above function Θ(n 12 logn) ? If no, explain very clearly why it is not (it cannot be). If yes, show a detailed step-by-sten proof of that in a similar fashion to what was done in \# i above

Answer 1

To prove or disprove the function's growth rate, we analyze the terms and compare them to the desired growth rate.

Step-by-step explanation:

To prove or disprove that f(n) = 5n^12 + 4n^9 log(n) + 36n^4 + 20n^3 + 50n + 50 is Ω(n^9 log(n)), we need to find positive constants c and k such that for all n ≥ k, f(n) ≥ c * n^9 log(n).

First, let's analyze the terms of f(n). The dominant term is 5n^12, which grows faster than n^9 log(n). Therefore, we can conclude that f(n) is not Ω(n^9 log(n)).

To determine whether f(n) is Θ(n^12 log(n)), we need to find positive constants c1, c2, and k such that for all n ≥ k, c1 * n^12 log(n) ≤ f(n) ≤ c2 * n^12 log(n).

Since the dominant term of f(n) is 5n^12, we can't find constants c1 and k that satisfy the first inequality. Therefore, we can conclude that f(n) is not Θ(n^12 log(n)).