The image presents the following problem:

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?

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:

• $C_3 = 2C_1 + 7C_2$
• $n_3 = \max(n_1, n_2)$

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.