Given \(f^{\prime}(x)=3(x^{2}+2x)-\frac{4}{x^{3}}+5\), we need to find \(f(x)\) such that \(f(1)=0\).
First, integrate \(f^{\prime}(x)\) with respect to \(x\):
\(f(x) = \int f^{\prime}(x) \, dx = \int \left(3x^{2}+6x-\frac{4}{x^{3}}+5\right) \, dx\)
\(f(x) = \int (3x^2 + 6x - 4x^{-3} + 5) \, dx\)
\(f(x) = 3\int x^2 \, dx + 6\int x \, dx - 4\int x^{-3} \, dx + 5\int dx\)
\(f(x) = 3\left(\frac{x^3}{3}\right) + 6\left(\frac{x^2}{2}\right) - 4\left(\frac{x^{-2}}{-2}\right) + 5x + C\)
\(f(x) = x^3 + 3x^2 + \frac{2}{x^2} + 5x + C\)
Now, use the initial condition \(f(1) = 0\) to find the constant \(C\):
\(0 = (1)^3 + 3(1)^2 + \frac{2}{(1)^2} + 5(1) + C\)
\(0 = 1 + 3 + 2 + 5 + C\)
\(0 = 11 + C\)
\(C = -11\)
Substitute the value of \(C\) back into the expression for \(f(x)\):
\(f(x) = x^3 + 3x^2 + \frac{2}{x^2} + 5x - 11\)
Correct Answer: \(x^{3}+3x^{2}+\frac{2}{x^{2}}+5x-11\)
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