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