https://file.aiquickdraw.com/mathbot/file/1727050322055nexvah8j.jpg

The problem in the image asks us to calculate \( f''(2) \) for the function:

\[
f(x) = x^3 + e^x(x^2 + x + 1)
\]

### Steps to solve this:

1. **Differentiate \( f(x) \) to find \( f'(x) \):**
   - Apply the product rule to the second term \( e^x(x^2 + x + 1) \), as it is a product of two functions.
   - The derivative of \( x^3 \) is straightforward.

2. **Differentiate again to find \( f''(x) \).**

3. **Substitute \( x = 2 \) into \( f''(x) \) to calculate \( f''(2) \).**

Let's start with the differentiation. I'll compute this now.The second derivative \( f''(x) \) is:

\[
f''(x) = 6x + 2(2x + 1)e^x + (x^2 + x + 1)e^x + 2e^x
\]

Now, substituting \( x = 2 \), we get:

\[
f''(2) = 12 + 19e^2
\]

This is the exact expression for \( f''(2) \).

If you need a more detailed explanation or have any further questions, feel free to ask! Here are some related questions:

1. How does the product rule work in calculus?
2. Why is the second derivative important in analyzing functions?
3. What are some common applications of second derivatives in real-world problems?
4. How do exponential functions affect the behavior of derivatives?
5. Can we approximate \( e^2 \) to get a numerical value for \( f''(2) \)?

**Tip:** When working with exponential functions, it's often helpful to keep results in symbolic form to avoid approximation errors until the final step.

Calculate f''(2) for f(x) = x^3 + e^x(x^2 + x + 1)

Learn how to calculate the second derivative f''(2) for the function f(x) = x^3 + e^x(x^2 + x + 1). This problem covers advanced calculus topics such as the product rule, chain rule, and exponential derivatives. Suitable for undergraduate calculus students.

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