Answer :
Final answer:
To find f(x) given f'(x) = 1 + 3√(x) and f(9) = 71, we integrate f'(x) and use the given condition to solve for the constant of integration, leading to f(x) = x + 2x√(x) + 8.
Explanation:
The question asks to find f(x) given that f'(x) = 1 + 3√(x), and f(9) = 71. The process involves integration of the derivative f'(x) to find f(x) and then using the initial condition to solve for the constant of integration.
To integrate f'(x), we get:
- ∫(1 + 3√(x))dx
- = ∫ dx + 3∫√(x) dx
- = x + 2x√(x) + C, where C is the constant of integration.
Using the initial condition f(9) = 71, we find:
- 9 + 2(9√(9)) + C = 71
- 9 + 2(27) + C = 71
- C = 8
Therefore, f(x) = x + 2x√(x) + 8.
