Answer is: option2

\( y = x^2 - \frac{1}{x} + 3 \)

Solution:

Steps:

1. Differentiate each option.
2. Check which derivative matches \( \frac{dy}{dx} = 2x + x^{-2} \).
3. Verify that the correct equation also satisfies \( f(1) = 3 \).

Option A: \( y = 2x^2 + \frac{1}{x} \) \[ \frac{dy}{dx} = \frac{d}{dx} \left(2x^2 + x^{-1} \right) \] \[ = 4x - x^{-2} \]

This is not equal to \( 2x + x^{-2} \), so Option A is incorrect.

Option B: \( y = x^2 - \frac{1}{x} + 3 \) \[ \frac{dy}{dx} = \frac{d}{dx} \left( x^2 - x^{-1} + 3 \right) \] \[ = 2x + x^{-2} \]

This matches the given derivative. So Option B is a possible correct answer.

Verifying \( f(1) = 3 \) for Option B:

We substitute \( x = 1 \) into the correct equation from Step 2, which is Option B:

\[ y = 1^2 - \frac{1}{1} + 3 \] \[ = 1 - 1 + 3 \] \[ = 3 \]

Since this satisfies the given point \( (1,3) \), Option B is the correct answer.