The derivative of a function f with respect to x is given by \(f'(x) = 3x^{2} - \frac{4}{x^{5}}\). If \(f(1) = 4\), find f(x).

The derivative of a function f with respect to x is given by \(f'(x) = 3x^{2} - \frac{4}{x^{5}}\). If \(f(1) = 4\), find f(x).

Answer Details

To find the function f(x) from its derivative, we need to integrate the derivative with respect to x. \[f(x) = \int f'(x) dx = \int (3x^{2} - \frac{4}{x^{5}}) dx = x^{3} + \frac{1}{x^{4}} + C\] where C is the constant of integration. We can use the given information that f(1) = 4 to find the value of C: \[f(1) = 1 + 1 + C = 4\] \[C = 2\] Substituting this value of C, we get: \[f(x) = x^{3} + \frac{1}{x^{4}} + 2\] Therefore, the answer is option (B) \(f(x) = x^{3} + \frac{1}{x^{4}} + 2\).