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Tjek om denne opgave er rigtig besvaret. De endelige svar er angivet med gul markering

Let's go through the problem step by step to verify if the answers are correct.

We are given the function:
\[
f(x) = ax^3 + 5x^2 + 2x + 1
\]
where \(a\) is a constant, and we need to:

### a) Find \( f'(x) \)

The derivative \( f'(x) \) can be calculated as follows:

1. The derivative of \( ax^3 \) is \( 3ax^2 \) (using the power rule).
2. The derivative of \( 5x^2 \) is \( 10x \).
3. The derivative of \( 2x \) is \( 2 \).
4. The derivative of the constant \( 1 \) is \( 0 \).

So:
\[
f'(x) = 3ax^2 + 10x + 2
\]
This matches the answer highlighted in yellow for part **a)**, so the first part is correct.

### b) Find \( a \) such that \( f'(1) = -3 \)

To solve for \(a\), substitute \(x = 1\) into the derivative equation \( f'(x) = 3ax^2 + 10x + 2 \):

\[
f'(1) = 3a(1)^2 + 10(1) + 2 = 3a + 10 + 2
\]
Simplifying:
\[
f'(1) = 3a + 12
\]
We are told that \( f'(1) = -3 \), so:
\[
3a + 12 = -3
\]
Solving for \(a\):
\[
3a = -3 - 12
\]
\[
3a = -15
\]
\[
a = \frac{-15}{3} = -5
\]
This calculation is correctly shown in the solution as well.

Finally, to verify the value of \( f'(1) \):
\[
f'(1) = 3(-5) + 12 = -15 + 12 = -3
\]
This confirms that the derivative at \( x = 1 \) is indeed \( -3 \).

### Conclusion:
Both parts **a)** and **b)** are correctly solved.

---

Do you have any more questions, or would you like a detailed explanation of a particular step?

Here are five related questions you might find useful:
1. How do you derive the derivative of a polynomial function?
2. Why is the derivative of a constant always zero?
3. How do you apply the power rule to functions with higher powers of \(x\)?
4. Can this approach be extended to functions involving fractional exponents?
5. How does changing the value of \(a\) affect the shape of the function?

**Tip**: When solving for constants in derivative problems, always substitute carefully and check your work by re-evaluating the function.

En funktion f er givet ved f(x) = ax^3 + 5x^2 + 2x + 1, hvor a er en konstant. a) Bestem f'(x). b) Bestem a, så f'(1) = -3.

This problem focuses on finding the derivative of a polynomial function f(x) = ax^3 + 5x^2 + 2x + 1 and solving for a constant a such that f'(1) = -3. Step-by-step solutions are provided for differentiation and solving for a.

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