For each of these functions, find the least integer n such that f(x) is O(x^n). 

If f(x) = 2x^2 + x^3logx, then the least integer n is:

To determine the least integer \( n \) such that \( f(x) = 2x^2 + x^3 \log x \) is \( O(x^n) \), we need to focus on the asymptotic growth of the function's terms as \( x \to \infty \).

### Step-by-Step Process:
1. **Break the function into parts:**
   - \( f(x) = 2x^2 + x^3 \log x \)
   - Here we have two terms: \( 2x^2 \) and \( x^3 \log x \).

2. **Analyze the growth of each term:**
   - \( 2x^2 \): This term grows like \( x^2 \).
   - \( x^3 \log x \): This term grows faster than \( x^3 \), but slightly slower than \( x^4 \), due to the logarithmic factor.

3. **Find the dominant term:**
   - Between \( 2x^2 \) and \( x^3 \log x \), the term \( x^3 \log x \) grows faster than \( 2x^2 \) as \( x \to \infty \).
   - Therefore, the term \( x^3 \log x \) dominates the growth of \( f(x) \).

4. **Estimate the order of the function:**
   - The logarithmic term \( \log x \) grows slower than any power of \( x \), but \( x^3 \log x \) grows slightly faster than \( x^3 \).
   - Since we are interested in the least integer \( n \) such that \( f(x) = O(x^n) \), we should approximate the growth of \( x^3 \log x \) as \( O(x^3) \), because \( \log x \) grows very slowly compared to powers of \( x \).

### Conclusion:
Thus, the least integer \( n \) such that \( f(x) = O(x^n) \) is \( \boxed{3} \).

Would you like further clarification on any step or concept? Here are some related questions to explore:

1. What is Big-O notation, and how is it used to estimate the growth of functions?
2. How does the logarithmic function compare to polynomial functions in terms of growth?
3. Why does the term \( x^3 \log x \) dominate over \( 2x^2 \) for large values of \( x \)?
4. What is the difference between Big-O and little-o notation?
5. How would the result change if \( f(x) \) included an exponential term?

**Tip:** When comparing terms for Big-O, always focus on the highest power of \( x \) and consider additional slow-growing factors like logarithms only after comparing the powers.

For each of these functions, find the least integer n such that f(x) is O(x^n). If f(x) = 2x^2 + x^3 logx, then the least integer n is:

Discover how to find the least integer n such that f(x) = 2x^2 + x^3 logx is O(x^n). This solution explains the use of Big-O notation and analyzes the asymptotic growth of polynomial and logarithmic functions. Suitable for college-level students.

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