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This problem involves analyzing the function \( f(x) \) defined as follows:

\[
f(x) = \frac{5x + 10}{e^{x^2 + 3}}
\]

We are tasked with two parts:

### Part a:
Verify that for all real \( x \) in the interval \([0, 3]\), the derivative of \( f(x) \) is given by:

\[
f'(x) = \frac{-10x^2 - 20x + 5}{e^{x^2 + 3}}
\]

### Part b:
Show that \( f(x) \) admits a maximum on the interval \([0, 3]\).

---

Let's first address **part a** by calculating the derivative \( f'(x) \). The function \( f(x) = \frac{5x + 10}{e^{x^2 + 3}} \) is a quotient, so we will apply the **quotient rule**:

\[
f'(x) = \frac{(u'(x) \cdot v(x) - u(x) \cdot v'(x))}{(v(x))^2}
\]

Where:
- \( u(x) = 5x + 10 \) and \( u'(x) = 5 \)
- \( v(x) = e^{x^2 + 3} \) and \( v'(x) = e^{x^2 + 3} \cdot 2x \) (using the chain rule)

Substituting these into the quotient rule, we can verify the derivative.

For **part b**, we will analyze \( f'(x) \) to find critical points and use the second derivative test or examine endpoints to show that \( f(x) \) admits a maximum on \([0, 3]\).

I will now calculate the derivative explicitly and check if the given \( f'(x) \) is correct. Let me do that now.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through the manual steps to solve this derivative, or feel free to ask about any other questions you might have! 

Here are some related questions you may consider:
1. What are the conditions for applying the quotient rule?
2. How do we apply the chain rule in functions involving exponentials?
3. What is the significance of critical points in determining maxima and minima?
4. How does the second derivative test work for identifying maximum or minimum points?
5. How can we use boundary values to determine extrema on a closed interval?

**Tip**: Always verify your derivative calculations by simplifying the result and checking consistency with initial conditions!

Verify that for all real x in the interval [0, 3], the derivative of the function f(x) = (5x + 10) / e^(x^2 + 3) is given by f'(x) = (-10x^2 - 20x + 5) / e^(x^2 + 3), and show that f admits a maximum on the interval [0, 3].

This problem involves applying the quotient rule to verify the derivative of a function and using critical points to show that the function admits a maximum on the interval [0, 3]. Concepts covered include the quotient rule, chain rule, and critical points analysis. Suitable for advanced high school or first-year calculus students.

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