Question

1. What is the best asymptotic ("big-O”) characterization of the following function: f(n) = (14logn)2 + log (3) a) 0(3) Show steps. b) O(n) c) 0(n) d) 0(21) e) O(logn)

Answer 1

Answer

$f(n)=(14\log{n})^2+\log(3^n)$

$f(n)=(14\log{n})^2+n\log(3)$

$f(n)=(14\log{n})^2+n(1.6)$

$\color{red}c)\;f(n)=O(n)$

Proof:

f(n) = O(g(n)) means there are positive constants c and n0, such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0

(14logn)^2 + 1.6n = O(n)
=>  (14logn)^2 + 1.6n <= O(n)
Let's assume c = 2
=>  (14logn)^2 + 1.6n <= O(n)
=>  (14logn)^2 + 1.6n <= 2(n)
=>  (14logn)^2 <= 0.4(n)
=>  2.5(14logn)^2 <= (n)
it's true for all n >= 100

so, (14logn)^2 + 1.6n = O(n) given c = 2 and n0 = 100, using the definition of Big-O