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.