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To solve this problem, we need to find the function \( f(x) \) given its derivative \( f'(x) = 2x^2 + 5x - 4 \) and the initial condition \( f(0) = 2 \).

### Step 1: Integrate \( f'(x) \)

The function \( f(x) \) can be found by integrating its derivative:

\[
f(x) = \int (2x^2 + 5x - 4) \, dx
\]

Integrating term-by-term, we get:

\[
f(x) = \frac{2x^3}{3} + \frac{5x^2}{2} - 4x + C
\]

where \( C \) is the constant of integration.

### Step 2: Apply the Initial Condition

We know that \( f(0) = 2 \). Substituting \( x = 0 \) into the function:

\[
f(0) = \frac{2(0)^3}{3} + \frac{5(0)^2}{2} - 4(0) + C = 2
\]

Simplifying, we find:

\[
C = 2
\]

### Step 3: Write the Final Function

Substitute the value of \( C \) back into the expression for \( f(x) \):

\[
f(x) = \frac{2x^3}{3} + \frac{5x^2}{2} - 4x + 2
\]

### Final Answer

The function \( f(x) \) is:

\[
\boxed{f(x) = \frac{2x^3}{3} + \frac{5x^2}{2} - 4x + 2}
\]

Would you like more details on the integration process or any other part of the solution?

### Further Related Questions

1. What is the second derivative \( f''(x) \) of the function \( f(x) \)?
2. How does the graph of \( f(x) \) behave, and where are its critical points?
3. What is the indefinite integral of \( f(x) \) itself?
4. How does the initial condition affect the solution for \( f(x) \)?
5. What is the area under the curve of \( f'(x) \) from \( x = 0 \) to \( x = 1 \)?

#### Tip:

When integrating a polynomial function, remember to increase the power by one and divide by the new power!

Find f such that f'(x) = 2x^2 + 5x - 4 and f(0) = 2.

Learn how to find a function f(x) from its derivative f'(x) = 2x^2 + 5x - 4 using integration and an initial condition f(0) = 2. This problem involves key calculus concepts and is suitable for students in Grades 11-12.

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