An object is moving along the `x`axis with postion as a function of time given by `x =x(t)`. Point `O` is at `x = 0`. Object is definitely moving towardr `O` when

Options

A. `dx//dt lt0`
B. `dx//dt gt 0`
C. `dx^(2)//dt^(2) lt 0`
D. `dx^(2)//dt^(2) gt 0`

Explanation

To determine when an object is moving towards point O (the origin at \(x = 0\)), we need to analyze its position function \(x(t)\) and its velocity \(v(t)\).

### Step-by-Step Solution:

1. **Understanding the Position Function**:
The position of the object is given by \(x(t)\), which indicates its location along the x-axis at any time \(t\).

2. **Defining Movement Towards the Origin**:
The object is moving towards the origin if its position \(x(t)\) is decreasing over time, meaning it is getting closer to \(x = 0\).

3. **Velocity Calculation**:
The velocity \(v(t)\) of the object is the derivative of the position function with respect to time:
\[ v(t) = \frac{dx}{dt} \]

4. **Condition for Moving Towards the Origin**:
For the object to be moving towards the origin, the following condition must hold:
- If \(x(t) > 0\) (the object is to the right of the origin), then \(v(t) < 0\) (the object must be moving left).
- If \(x(t) < 0\) (the object is to the left of the origin), then \(v(t) > 0\) (the object must be moving right).

5. **Combining Conditions**:
We can express this mathematically:
\[ x(t) \cdot v(t) < 0 \]
This means that the position \(x(t)\) and the velocity \(v(t)\) must have opposite signs for the object to be moving towards the origin.

6. **Conclusion**:
Therefore, the object is definitely moving towards point O when the product of its position and velocity is negative:
\[ x(t) \cdot \frac{dx}{dt} < 0 \]