Step 1: Concept Overview:
Given the derivative \(f'(x)\) and a point \(f(1)=0\), we aim to determine the original function \(f(x)\). This involves integrating \(f'(x)\) and using the initial condition to solve for the integration constant, \(C\).
Step 2: Methodology:
1. Integrate \(f'(x)\) to obtain \(f(x)\): \(f(x) = \int f'(x) dx\).
2. Apply the power rule: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).
3. Employ the condition \(f(1)=0\) to find \(C\).
Step 3: Detailed Solution:
We have \(f'(x) = 3x^2 - \frac{2}{x^2}\), which can be written as \(f'(x) = 3x^2 - 2x^{-2}\).
Integrate \(f'(x)\) to get \(f(x)\):\[ f(x) = \int (3x^2 - 2x^{-2}) dx \]Applying the power rule to each term:\[ f(x) = 3 \int x^2 dx - 2 \int x^{-2} dx \]\[ f(x) = 3 \left(\frac{x^{2+1}}{2+1}\right) - 2 \left(\frac{x^{-2+1}}{-2+1}\right) + C \]\[ f(x) = 3 \left(\frac{x^3}{3}\right) - 2 \left(\frac{x^{-1}}{-1}\right) + C \]\[ f(x) = x^3 + 2x^{-1} + C \]\[ f(x) = x^3 + \frac{2}{x} + C \]Using the condition \(f(1) = 0\) to determine \(C\):\[ f(1) = (1)^3 + \frac{2}{1} + C = 0 \]\[ 1 + 2 + C = 0 \]\[ 3 + C = 0 \]\[ C = -3 \]Substitute \(C\) back into \(f(x)\):\[ f(x) = x^3 + \frac{2}{x} - 3 \]
Step 4: Final Result:
The function is \(f(x) = x^3 + \frac{2}{x} - 3\), corresponding to option (C).