https://r2.erweima.ai/mathbot/image/1726592209162lu719yo0.jpg

The function provided in the image is:

\[
f(x) = \frac{3x^3 + x + 2}{5x^2 + 1}
\]

Would you like me to:

1. Find the derivative of the function \( f'(x) \) using the quotient rule?
2. Analyze the critical points and determine where the function is increasing or decreasing?
3. Discuss the asymptotic behavior or limits as \( x \to \infty \) or \( x \to -\infty \)?
4. Find any vertical or horizontal asymptotes?
5. Plot or sketch the function for a visual representation?

Let me know if you'd like any details on these, or have a different request!

### Tip:
When dealing with rational functions, always check the denominator for points where the function might be undefined.

This problem involves differentiating and analyzing the rational function f(x) = (3x^3 + x + 2) / (5x^2 + 1). Learn how to apply the quotient rule to find the derivative and analyze the function's behavior. Suitable for Grades 11-12 students studying calculus.

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