Question

If \(f'(x) = 3x^2 - \frac{2}{x^2}\), \(f(1) = 0\) then, \(f(x)\) is

• A. \(x^3 + \frac{2}{x^2} - 3\)
• B. \(x^3 + \frac{1}{x^2} + 3\)
• C. \(x^3 + \frac{2}{x} - 3\)
• D. \(x^3 + \frac{2}{x^2} + 3\)

Hint:

Be careful with negative exponents during integration. The integral of \(x^{-2}\) is \(\frac{x^{-1}}{-1} = -\frac{1}{x}\), not \(\frac{x^{-3}}{-3}\). Always double-check your application of the power rule, especially with signs.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are given the derivative of a function, \(f'(x)\), and a point on the function, \(f(1)=0\). To find the original function \(f(x)\), we need to integrate the derivative \(f'(x)\) and then use the given point (initial condition) to find the constant of integration, \(C\).

Step 2: Key Formula or Approach:
1. Find the indefinite integral of \(f'(x)\) to get \(f(x)\): \(f(x) = \int f'(x) dx\).
2. Use the power rule for integration: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).
3. Use the given condition \(f(1)=0\) to solve for the constant of integration \(C\).

Step 3: Detailed Explanation:
We are given \(f'(x) = 3x^2 - \frac{2}{x^2}\). We can rewrite this as \(f'(x) = 3x^2 - 2x^{-2}\).
Now, we integrate \(f'(x)\) to find \(f(x)\): \[ f(x) = \int (3x^2 - 2x^{-2}) dx \] Applying the power rule for integration to each term: \[ f(x) = 3 \int x^2 dx - 2 \int x^{-2} dx \] \[ f(x) = 3 \left(\frac{x^{2+1}}{2+1}\right) - 2 \left(\frac{x^{-2+1}}{-2+1}\right) + C \] \[ f(x) = 3 \left(\frac{x^3}{3}\right) - 2 \left(\frac{x^{-1}}{-1}\right) + C \] \[ f(x) = x^3 + 2x^{-1} + C \] \[ f(x) = x^3 + \frac{2}{x} + C \] Now, we use the initial condition \(f(1) = 0\) to find the value of \(C\): \[ f(1) = (1)^3 + \frac{2}{1} + C = 0 \] \[ 1 + 2 + C = 0 \] \[ 3 + C = 0 \] \[ C = -3 \] Substituting the value of \(C\) back into the expression for \(f(x)\): \[ f(x) = x^3 + \frac{2}{x} - 3 \]
Step 4: Final Answer:
The function is \(f(x) = x^3 + \frac{2}{x} - 3\), which corresponds to option (C).