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: B
Explanation
\(\frac{\mathrm d y}{\mathrm d x} = 3x^{2} - \frac{4}{x^5} = 3x^{2} - 4x^{-5}\)
\(y = \int (3x^{2} - 4x^{-5}) \mathrm {d} x \)
\(y = x^{3} + \frac{1}{x^{4}} + c\)
f(1) = 4; \(4 = 1^{3} + \frac{1}{1^{4}} + c \implies 4 = 2 + c\)
\(c = 2\)
\(f(x) = x^{3} + \frac{1}{x^{4}} + 2\)
