2 Asymptotic Notation - Bounds
Problem: Let f (n) = n4 + 6n2 + 5. Prove that, for all integers numbers k, we have that f(n) = O(nk) if and only if k ≥ 4.
Hint: Because of the phrasing “if and only if” we must prove that f(n) = O(nk) when k ≥ 4 and prove that f(n) ̸= O(nk) when k < 4.
Hint: Focus on the cases of k = 4 and k = 3. How can the arguments for these cases be extended to all other k?

To solve this asymptotic notation problem, we need to prove the following statement:

**Statement:**  
Let \( f(n) = n^4 + 6n^2 + 5 \). Then, for all integers \( k \), \( f(n) = O(n^k) \) if and only if \( k \geq 4 \).

This is an "if and only if" statement, which means we need to prove it in two parts:
1. **(If)**: Show that \( f(n) = O(n^k) \) when \( k \geq 4 \).
2. **(Only if)**: Show that \( f(n) \neq O(n^k) \) when \( k < 4 \).

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### Step 1: Prove \( f(n) = O(n^k) \) when \( k \geq 4 \)
We start by analyzing the function \( f(n) = n^4 + 6n^2 + 5 \).

#### Case 1: \( k = 4 \)
To show that \( f(n) = O(n^4) \), we need to find constants \( c > 0 \) and \( n_0 \geq 1 \) such that for all \( n \geq n_0 \):
\[
f(n) \leq c n^4
\]

We have:
\[
f(n) = n^4 + 6n^2 + 5
\]
For large \( n \), the dominant term in \( f(n) \) is \( n^4 \), while the other terms \( 6n^2 \) and 5 are smaller in comparison.

For sufficiently large \( n \), we know that:
\[
n^4 + 6n^2 + 5 \leq n^4 + n^4 + n^4 = 3n^4
\]
Thus, for large \( n \), we can choose \( c = 3 \) and \( n_0 = 1 \), so:
\[
f(n) \leq 3n^4 \quad \text{for all } n \geq 1
\]
Therefore, \( f(n) = O(n^4) \).

#### Case 2: \( k > 4 \)
When \( k > 4 \), we need to show that \( f(n) = O(n^k) \). Since \( n^k \) grows faster than \( n^4 \) for large \( n \), we know that:
\[
f(n) = n^4 + 6n^2 + 5 \leq n^k \quad \text{for sufficiently large } n
\]
For large \( n \), \( n^4 \) is much smaller than \( n^k \). So, for sufficiently large \( n \), the inequality holds. Thus, \( f(n) = O(n^k) \) when \( k > 4 \).

---

### Step 2: Prove \( f(n) \neq O(n^k) \) when \( k < 4 \)
Now, we need to show that \( f(n) \neq O(n^k) \) when \( k < 4 \). The main idea here is that the highest degree term in \( f(n) \) is \( n^4 \), and if we try to bound it by \( n^k \) for \( k < 4 \), the bound will not hold for large \( n \).

#### Case 1: \( k = 3 \)
Suppose \( f(n) = O(n^3) \). This would mean there exists a constant \( c > 0 \) and \( n_0 \geq 1 \) such that:
\[
f(n) \leq c n^3 \quad \text{for all } n \geq n_0
\]
However, for large \( n \), the dominant term in \( f(n) \) is \( n^4 \), and we have:
\[
f(n) = n^4 + 6n^2 + 5
\]
Since \( n^4 \) grows faster than \( n^3 \), it is impossible to find such a constant \( c \) such that \( n^4 \leq c n^3 \) for large \( n \). Therefore, \( f(n) \neq O(n^3) \).

#### Case 2: \( k < 3 \)
If \( k < 3 \), the argument is even stronger. The term \( n^4 \) will dominate even more significantly, and no constant \( c \) will satisfy \( f(n) \leq c n^k \) for large \( n \). Hence, \( f(n) \neq O(n^k) \) when \( k < 4 \).

---

### Conclusion
- We have shown that \( f(n) = O(n^k) \) for \( k \geq 4 \).
- We have also shown that \( f(n) \neq O(n^k) \) for \( k < 4 \).

Thus, \( f(n) = O(n^k) \) if and only if \( k \geq 4 \), completing the proof.

---

Do you want further clarifications on any step or details? Here are five related questions you might consider:

1. How does the behavior of asymptotic notation change when \( f(n) \) has logarithmic terms?
2. How can we use limits to formally define Big-O and prove the bounds?
3. How would the proof change if \( f(n) \) had higher-degree terms like \( n^5 \)?
4. Can we provide a visual comparison of \( f(n) \) with \( n^k \) for different \( k \)?
5. What are the common misunderstandings with asymptotic notation in algorithm analysis?

**Tip:** When using Big-O notation, always focus on the term with the highest growth rate, as it dominates the behavior of the function for large \( n \).

Let f(n) = n^4 + 6n^2 + 5. Prove that, for all integers k, we have that f(n) = O(n^k) if and only if k ≥ 4.

Learn how to prove that the polynomial function f(n) = n^4 + 6n^2 + 5 is O(n^k) if and only if k ≥ 4. This problem demonstrates the application of Big-O notation, focusing on polynomial growth and asymptotic analysis. Suitable for university-level students in algorithms or computer science courses.

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