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!