Question

Let f(n)= 14n³/²+ 12n³ + 10²⁰. Replace the question mark with

the correct function
(tightest bound) for each case below. Justify your answer clearly
for each case.
a) f(n)=O(?)
b) f(n)=Ω(?)
c) f(n)=theta(?)

Answer 1

Let f(n)= 14n³/²+ 12n³ + 10²⁰. Replace the question mark with a) f(n)=O(n³) b) f(n)=Ω(n³) c) f(n)=θ(n³)

Step-by-step explanation:

The tightest bound for the given function f(n) = 14n^(3/2) + 12n³ + 10²⁰ can be determined by analyzing its growth rate. When considering the term with the highest growth rate, which is 14n^(3/2), it becomes evident that this term dominates the overall growth of the function. Therefore, the function can be bounded above by O(n³), as n^(3/2) grows slower than n³. This means that there exists a constant c and a value n₀ such that 14n^(3/2) + 12n³ + 10²⁰ ≤ cn³ for all n > n₀, thus justifying the notation f(n)=O(n³).

On the other hand, when analyzing the lower bound, it is clear that the term 12n³ dominates the overall growth of the function. Hence, the function can be bounded below by Ω(n³), as there exists a constant c and a value n₀ such that 14n^(3/2) + 12n³ + 10²⁰ ≥ cn³ for all n > n₀, justifying the notation f(n)=Ω(n³).

Finally, since both O(n³) and Ω(n³) are applicable to the given function, it follows that f(n)=θ(n³), indicating that the function grows at the same rate as n³.

In conclusion, through careful analysis of the growth rates of each term in the given function, it is evident that f(n) = 14n^(3/2) + 12n³ + 10²⁰ can be tightly bounded above and below by n³, leading to the conclusion that f(n)=θ(n³).

Correct option is a) f(n)=O(n³) b) f(n)=Ω(n³) c) f(n)=θ(n³)