(c) Show that f(n) = 3n3 + 7n + 10 ∈ θ(n3)

To show that the function \( f(n) = 3n^3 + 7n + 10 \) belongs to \( \theta(n^3) \), we need to demonstrate that there exist positive constants \( c_1 \), \( c_2 \), and \( n_0 \) such that for all \( n \geq n_0 \):

\[
c_1 n^3 \leq f(n) \leq c_2 n^3
\]

### Step 1: Upper Bound
We first show that \( f(n) \leq c_2 n^3 \) for some constant \( c_2 \) and sufficiently large \( n \).

\[
f(n) = 3n^3 + 7n + 10
\]

For large \( n \), the term \( 3n^3 \) will dominate, but we must include all terms for correctness:

\[
f(n) \leq 3n^3 + 7n^3 + 10n^3 = 3n^3 + \frac{7}{n^2} n^3 + \frac{10}{n^3} n^3
\]

Now for large enough \( n \), the terms \( \frac{7}{n^2} n^3 \) and \( \frac{10}{n^3} n^3 \) become negligible compared to \( 3n^3 \). However, let’s combine all terms under a single constant:

\[
f(n) \leq (3 + \epsilon) n^3 \quad \text{for some small } \epsilon > 0
\]

Therefore, for a sufficiently large \( n \), we can choose \( c_2 = 4 \) (since 3 + any small \(\epsilon\) will be less than 4):

\[
f(n) \leq 4n^3
\]

### Step 2: Lower Bound
We now show that \( f(n) \geq c_1 n^3 \) for some constant \( c_1 \) and sufficiently large \( n \).

Notice:

\[
f(n) = 3n^3 + 7n + 10 \geq 3n^3
\]

This holds for any \( n \geq 1 \), so we can take \( c_1 = 3 \).

### Step 3: Conclusion

We have shown that:

\[
3n^3 \leq f(n) \leq 4n^3
\]

Thus, \( f(n) = 3n^3 + 7n + 10 \) satisfies the definition of \( \theta(n^3) \), meaning:

\[
f(n) \in \theta(n^3)
\]

This concludes the proof.

---

Would you like to see more details or have any questions? Here are 5 related questions:

1. What are other examples of functions that belong to \( \theta(n^3) \)?
2. How does the concept of \( \theta \)-notation differ from \( O \)-notation?
3. Can you explain how to determine \( \theta \)-notation for a more complex function?
4. How would you show that a function belongs to \( \theta(n^2) \) instead?
5. What is the significance of asymptotic notations in algorithm analysis?

**Tip:** When proving that a function belongs to a particular asymptotic notation, always remember to analyze both the upper and lower bounds carefully.

Learn how to prove that the function f(n) = 3n^3 + 7n + 10 belongs to θ(n^3). This involves demonstrating both upper and lower bounds using asymptotic notation. Suitable for advanced mathematics students or those studying asymptotic analysis.

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