Question: Question number 14. Find \( f(x) \) given that \( f’(x) = 4x + 3 \) and \( f(0) = 7 \). -…

Question number 14.

Find \( f(x) \) given that \( f’(x) = 4x + 3 \) and \( f(0) = 7 \).

• \( f(x) = 4x + 10 \)
• \( f(x) = 2x^2 + 3x + 7 \)
• \( f(x) = 2x^2 + 3x \)
• \( f(x) = 7 \)
• \( f(x) = 4x + 3 \)

Question number 15.

Solution

To find \( f(x) \) given that \( f’(x) = 4x + 3 \) and \( f(0) = 7 \), we need to integrate \( f’(x) \). The integral of \( f’(x) = 4x + 3 \) is: \[ f(x) = \int (4x + 3) \, dx \] Integrating term by term, we get: \[ f(x) = \int 4x \, dx + \int 3 \, dx \] \[ f(x) = 2x^2 + 3x + C \] To find the constant \( C \), we use the initial condition \( f(0) = 7 \): \[ f(0) = 2(0)^2 + 3(0) + C = 7 \] \[ C = 7 \] Therefore, the function is: \[ f(x) = 2x^2 + 3x + 7 \] The correct answer is \( f(x) = 2x^2 + 3x + 7 \).