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The image presents the following problem:

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### Problem 5:

Suppose that:
- \( f(x) = \mathcal{O}(x^{n_1}) \) using constants \( C_1 \) and \( n_1 \geq 1 \).
- \( g(x) = \mathcal{O}(x^{n_2}) \) using constants \( C_2 \) and \( n_2 \geq 1 \).

We can prove that \( 2f(x) + 7g(x) = \mathcal{O}(x^{\max(n_1, n_2)}) \) using constants \( C_3 \) and \( n_3 \). Not knowing anything else, what are the minimum values of \( C_3 \) and \( n_3 \) which will prove this?

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### Solution approach:

We are asked to determine the asymptotic behavior of the sum \( 2f(x) + 7g(x) \) in Big-O notation.

1. **Step 1: Big-O representation of \( f(x) \) and \( g(x) \):**
   - By the problem, \( f(x) = \mathcal{O}(x^{n_1}) \), so there exist constants \( C_1 \) and \( n_1 \) such that \( |f(x)| \leq C_1 x^{n_1} \) for large \( x \).
   - Similarly, \( g(x) = \mathcal{O}(x^{n_2}) \), so there exist constants \( C_2 \) and \( n_2 \) such that \( |g(x)| \leq C_2 x^{n_2} \) for large \( x \).

2. **Step 2: Evaluate the sum \( 2f(x) + 7g(x) \):**
   - Using the properties of Big-O notation:
     \[
     |2f(x) + 7g(x)| \leq 2|f(x)| + 7|g(x)| \leq 2C_1 x^{n_1} + 7C_2 x^{n_2}.
     \]
   - Now, we need to determine the dominant term in the sum.

3. **Step 3: Maximum of powers:**
   - The dominant term will be the one with the largest power of \( x \), which is \( x^{\max(n_1, n_2)} \).

4. **Step 4: Conclusion:**
   - The sum \( 2f(x) + 7g(x) \) is bounded by \( \mathcal{O}(x^{\max(n_1, n_2)}) \).
   - The constant \( C_3 \) will be determined by the coefficients in front of \( f(x) \) and \( g(x) \), so we take \( C_3 = 2C_1 + 7C_2 \).

Thus, the minimum values of \( C_3 \) and \( n_3 \) are:
- \( C_3 = 2C_1 + 7C_2 \)
- \( n_3 = \max(n_1, n_2) \)

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Would you like more details on any part of this solution? Here are some related questions you might find useful:

1. How do we formally prove the properties of Big-O notation?
2. Can we generalize this approach to sums with more terms?
3. How does Big-O behave when multiplying functions instead of adding them?
4. What are the implications of choosing different constants in Big-O notation?
5. How does Big-O notation compare with other notations like Big-Omega and Big-Theta?

### Tip:
In Big-O problems, always focus on the dominant term, which determines the overall growth rate of the function for large inputs.

Suppose that:
- f(x) = O(x^n1) using constants C1 and n1 ≥ 1.
- g(x) = O(x^n2) using constants C2 and n2 ≥ 1.
We can prove that 2f(x) + 7g(x) = O(x^max(n1, n2)) using constants C3 and n3. Not knowing anything else, what are the minimum values of C3 and n3 which will prove this?

Explore the proof of asymptotic behavior for the sum 2f(x) + 7g(x) using Big-O notation. Learn how constants C3 and n3 are determined in relation to given functions f(x) and g(x) with constants C1 and C2, focusing on asymptotic dominance. Suitable for undergraduate students.

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