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The image contains a question related to integral calculus. The problem asks about determining which of the given functions \( f(x) \) satisfies the initial condition \( f(1) = 3 \) and the derivative \( f'(x) \).

Let's analyze the problem step by step.

### Given Data:
- We are given that the derivative \( f'(x) \) is \( f'(x) = 3x^2 - 5x + 2 \).
- We need to find the function \( f(x) \) that satisfies the initial condition \( f(1) = 3 \).

### Steps to Solve:

1. **Integrate \( f'(x) \) to Find \( f(x) \):**

   \[
   f(x) = \int (3x^2 - 5x + 2) \, dx
   \]

   Let's integrate each term:

   \[
   f(x) = \int 3x^2 \, dx - \int 5x \, dx + \int 2 \, dx
   \]

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

   Simplifying:

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

2. **Apply the Initial Condition \( f(1) = 3 \):**

   Substitute \( x = 1 \) into the equation:

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

   \[
   1 - \frac{5}{2} + 2 + C = 3
   \]

   \[
   1 - 2.5 + 2 + C = 3
   \]

   \[
   0.5 + C = 3
   \]

   \[
   C = 2.5
   \]

   Thus, the function is:

   \[
   f(x) = x^3 - \frac{5}{2}x^2 + 2x + 2.5
   \]

3. **Identify the Correct Option:**

   Based on the function derived, we check the options given in the image. The function we obtained is:

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

   Which corresponds to:

   \[
   \text{Option c) } f(x) = x^3 - \frac{5}{2}x^2 + 2x + \frac{5}{2}
   \]

### Final Answer:
The correct option is **c)**.

Would you like further details on any part of this solution? Here are five related questions you might find useful:

1. What is the general process of finding the antiderivative of a polynomial function?
2. How do you apply initial conditions to determine the constant of integration?
3. Why is the constant of integration important in indefinite integrals?
4. What are some common mistakes when solving indefinite integrals?
5. How does the Fundamental Theorem of Calculus connect derivatives and integrals?

**Tip:** Always double-check the initial conditions when solving problems involving indefinite integrals, as this ensures the correctness of the particular solution.

Given f'(x) = 3x^2 - 5x + 2 and f(1) = 3, find the correct f(x) among the options.

Learn how to solve the integral calculus problem where f'(x) = 3x^2 - 5x + 2 and f(1) = 3. Follow a step-by-step process to integrate f'(x) and apply the initial condition to determine the correct function f(x).

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