Question: Find \( f(x) \) given that \( f”(x) = 30x + 14 \), \( f(0) = 6 \), and \( f’(0) = -8 \)…

Find \( f(x) \) given that \( f”(x) = 30x + 14 \), \( f(0) = 6 \), and \( f’(0) = -8 \).

• \( f(x) = 5x^3 + 7x^2 - 8x + 6 \)
• \( f(x) = 5x^3 + 7x^2 + 6x + 6 \)
• \( f(x) = 5x^3 + 7x^2 + 6x - 8 \)
• \( f(x) = 15x^2 + 7x - 8 \)
• \( f(x) = 15x^2 + 7x + 6 \)
• \( f(x) = 5x^3 + 7x^2 - 8x - 8 \)

Solution

To find \( f(x) \), we need to integrate \( f”(x) = 30x + 14 \) twice. First, find the first derivative \( f’(x) \) by integrating \( f”(x) \): \[ f’(x) = \int (30x + 14) \, dx \] Calculate the integral: \[ f’(x) = 15x^2 + 14x + C \] where \( C \) is a constant. Given \( f’(0) = -8 \): \[ 15(0)^2 + 14(0) + C = -8 \] So, \( C = -8 \). Thus, the first derivative is: \[ f’(x) = 15x^2 + 14x - 8 \] Now, find \( f(x) \) by integrating \( f’(x) \): \[ f(x) = \int (15x^2 + 14x - 8) \, dx \] Calculate the integral: \[ f(x) = 5x^3 + 7x^2 - 8x + D \] where \( D \) is another constant. Given \( f(0) = 6 \): \[ 5(0)^3 + 7(0)^2 - 8(0) + D = 6 \] So, \( D = 6 \). Thus, \( f(x) = 5x^3 + 7x^2 - 8x + 6 \). The correct option is: \( f(x) = 5x^3 + 7x^2 - 8x + 6 \)