If `(d)/(dx)f(x) = 4x^(3) - (3)/(x^(4))` such that `f(2) = 0`. Then f(x) is

Options

A. `x^(4) + (1)/(x^(3)) - (129)/(8)`
B. `x^(3) + (1)/(x^(4)) + (129)/(8)`
C. `x^(4) + (1)/(x^(3)) + (129)/(8)`
D. `x^(3) + (1)/(x^(4)) - (129)/(8)`

Answer

To solve the problem, we need to find the function \( f(x) \) given that

\[ \frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4} \]

and that \( f(2) = 0 \).

### Step 1: Integrate the derivative

To find \( f(x) \), we need to integrate the expression \( 4x^3 - \frac{3}{x^4} \).

\[ f(x) = \int \left( 4x^3 - \frac{3}{x^4} \right) dx \]

### Step 2: Perform the integration

Now, we will integrate each term separately.

1. The integral of \( 4x^3 \):
\[ \int 4x^3 \, dx = 4 \cdot \frac{x^4}{4} = x^4 \]

2. The integral of \( -\frac{3}{x^4} \):
\[ \int -\frac{3}{x^4} \, dx = -3 \cdot \int x^{-4} \, dx = -3 \cdot \left( \frac{x^{-3}}{-3} \right) = \frac{1}{x^3} \]

Combining these results, we have:
\[ f(x) = x^4 + \frac{1}{x^3} + C \]

where \( C \) is the constant of integration.

### Step 3: Use the initial condition to find \( C \)

We know that \( f(2) = 0 \). We will substitute \( x = 2 \) into our equation to find \( C \).

\[ f(2) = 2^4 + \frac{1}{2^3} + C = 0 \]

Calculating \( f(2) \):
\[ f(2) = 16 + \frac{1}{8} + C = 0 \]
\[ 16 + 0.125 + C = 0 \]
\[ C = -16.125 \]

### Step 4: Write the final expression for \( f(x) \)

Now substituting \( C \) back into the equation for \( f(x) \):
\[ f(x) = x^4 + \frac{1}{x^3} - 16.125 \]

### Final Answer

Thus, the function \( f(x) \) is:
\[ f(x) = x^4 + \frac{1}{x^3} - \frac{129}{8} \]