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).

A.

\(f(x) = x^{3} - \frac{1}{x^{4}} + 2\)

B.

\(f(x) = x^{3} + \frac{1}{x^{4}} + 2\)

C.

\(f(x) = x^{3} - \frac{1}{x^{4}} - 2 \)

D.

\(f(x) = x^{3} + \frac{1}{x^{4}} - 2\)

Correct answer is B

\(\frac{\mathrm d y}{\mathrm d x} = 3x^{2} - \frac{4}{x^5} = 3x^{2} - 4x^{-5}\)

\(y = \int (3x^{2} - 4x^{-5}) \mathrm {d} x \)

\(y = x^{3} + \frac{1}{x^{4}} + c\)

f(1) = 4; \(4 = 1^{3} + \frac{1}{1^{4}} + c \implies 4 = 2 + c\)

\(c = 2\)

\(f(x) = x^{3} + \frac{1}{x^{4}} + 2\)