To find the function \( f(x) \) given its first derivative \( f'(x) = 3x + 2 \) and the fact that it passes through the origin, we need to follow a few steps involving integration and applying the initial condition. ### Step 1: Integrate the Derivative The first step is to integrate the derivative \( f'(x) \) to find the original function \( f(x) \). The derivative given is: \[ f'(x) = 3x + 2 \] To find \( f(x) \), we integrate \( f'(x) \): \[ f(x) = \int (3x + 2) \, dx \] ### Step 2: Perform the Integration Now, we perform the integration term by term: 1. The integral of \( 3x \) is \( \frac{3x^2}{2} \). 2. The integral of \( 2 \) is \( 2x \). Putting it all together, we have: \[ f(x) = \frac{3x^2}{2} + 2x + C \] where \( C \) is the constant of integration. ### Step 3: Apply the Initial Condition Since the function passes through the origin, we know that \( f(0) = 0 \). We can use this information to find the constant \( C \). Substituting \( x = 0 \) into the equation: \[ f(0) = \frac{3(0)^2}{2} + 2(0) + C = 0 \] This simplifies to: \[ C = 0 \] ### Step 4: Write the Final Function Now that we have determined \( C \), we can write the final form of the function: \[ f(x) = \frac{3x^2}{2} + 2x \] ### Step 5: Identify the Correct Option Now, let's compare this result with the provided options: - **A.** \( y = \frac{3x^2}{2} + 2x \) (This matches our derived function) - **B.** \( y = \frac{3x^2}{2} + x \) (This has an incorrect linear term) - **C.** \( y = 3x^2 + \frac{x}{2} \) (This has incorrect coefficients) - **D.** \( y = 3x^2 + 2x \) (This has an incorrect leading coefficient) The correct option is **A**. ### Explanation of Incorrect Options - **Option B**: The linear term is incorrect. It should be \( 2x \) instead of \( x \). - **Option C**: The coefficients of \( x^2 \) and \( x \) are incorrect. The function should have \( \frac{3}{2} \) as the coefficient of \( x^2 \) and \( 2 \) as the coefficient of \( x \). - **Option D**: The coefficient of \( x^2 \) is incorrect. It should be \( \frac{3}{2} \) instead of \( 3 \). ### Summary - To find \( f(x) \), integrate the derivative \( f'(x) = 3x + 2 \). - The integration yields \( f(x) = \frac{3x^2}{2} + 2x + C \). - Use the initial condition \( f(0) = 0 \) to find \( C = 0 \). - The final function is \( f(x) = \frac{3x^2}{2} + 2x \), which corresponds to option A. This thorough approach ensures a clear understanding of how to derive a function from its derivative and apply initial conditions correctly.