To solve this problem, we need to determine the smallest integer $n$ such that $f(x)$ is $O(x^n)$ for each function. This means we are looking for the asymptotic growth rate of each function $f(x)$ and identifying the term with the highest growth as $x \to \infty$.

Problem Analysis

1. Big-O Notation $O(x^n)$: This notation represents the upper bound of a function as $x$ grows large. To find the smallest $n$ such that $f(x) = O(x^n)$, we need to look at the term in $f(x)$ with the highest power of $x$ as $x \to \infty$, as this term dominates the growth of $f(x)$.

Solution for Each Function

Part (a)

Function: $f(x) = 2x^2 + x^6 \log(x)$

• The term $2x^2$ grows as $x^2$.
• The term $x^6 \log(x)$ grows faster because $x^6$ dominates $x^2$, and $\log(x)$ is a slowly increasing function.
• Thus, $x^6 \log(x)$ is the dominant term as $x \to \infty$.

Since $x^6 \log(x) = O(x^6)$, the smallest integer $n$ for which $f(x) = O(x^n)$ is $n = 6$.

Part (b)

Function: $f(x) = 3x^7 + \log(x)$

• The term $3x^7$ grows as $x^7$.
• The term $\log(x)$ grows slower than any power of $x$.
• Therefore, $3x^7$ is the dominant term as $x \to \infty$.

Since $3x^7 = O(x^7)$, the smallest integer $n$ for which $f(x) = O(x^n)$ is $n = 7$.

Part (c)

Function: $f(x) = \frac{x^4 + x^2 + 1}{x^4 + 1}$

• Divide the numerator and denominator by $x^4$ to simplify: $f(x) = \frac{1 + \frac{x^2}{x^4} + \frac{1}{x^4}}{1 + \frac{1}{x^4}} = \frac{1 + \frac{1}{x^2} + \frac{1}{x^4}}{1 + \frac{1}{x^4}}$
• As $x \to \infty$, the terms $\frac{1}{x^2}$ and $\frac{1}{x^4}$ in both the numerator and the denominator approach zero. Thus, $f(x)$ approaches $1$ as $x \to \infty$.
• This implies that $f(x)$ is asymptotically constant, so $f(x) = O(x^0)$.

Therefore, the smallest integer $n$ for which $f(x) = O(x^n)$ is $n = 0$.

Summary of Answers

• (a) $n = 6$
• (b) $n = 7$
• (c) $n = 0$

Would you like further details on any part of this solution or have any questions?

Related Questions:

1. How does Big-O notation help in determining the growth rate of functions?
2. What is the effect of logarithmic terms on the asymptotic growth of functions?
3. How would the dominant term change if the exponent of $x$ was altered in part (a)?
4. Why is $O(x^0)$ appropriate for constants?
5. How do polynomial terms dominate other terms in functions as $x \to \infty$?

Tip: When working with Big-O notation, always look for the term with the highest growth rate since it dominates the function's behavior as $x \to \infty$.