Integrating the Derivative to Find

The problem asks us to find the function $ f(x) $ given its derivative, $ f'(x) = 3x^2 - \frac{2}{x^2} $, and a specific value of the function, $ f(1) = 0 $. This requires us to perform integration and then use the given condition to find the constant of integration.

Step 1: Integrate $f'(x)$

First, we find the indefinite integral of $ f'(x) $ to get the general form of $ f(x) $. We can rewrite $ f'(x) $ as $ f'(x) = 3x^2 - 2x^{-2} $.

The integral is calculated as follows:

Using the power rule for integration, which states that $ \int ax^n dx = a \frac{x^{n+1}}{n+1} + C $ (where $ n \neq -1 $):

And for the second term:

Combining these results and adding the constant of integration $ C $:

Step 2: Use the Condition $f(1) = 0$

We are given that $ f(1) = 0 $. We substitute $ x = 1 $ into the expression for $ f(x) $ we found:

Since $ f(1) = 0 $, we set the expression equal to 0:

Solving for $ C $:

Step 3: Determine the Final Function $f(x)$

Now, we substitute the value $ C = -3 $ back into the general form of $ f(x) $:

This is the specific function that satisfies both the derivative $ f'(x) $ and the condition $ f(1) = 0 $.