If `y=x+[x],` then
Statement I `(dy)/(dx)=1` for all `x"inR`
Statement II`(d([x]))/(dx)={(" "0","" "x cancelin "Integer"),("does not exist"" "x.in"integer"`):}

Options

A. Both statement I and Statement II are correct and Statement II is the correct explanation of Statement I
B. Both Statement I and Statement II are correct but Statement II is not the correct explanation of Statement I
C. Statement I is correct but Statement II is incorrect
D. Statement II is correct but Statement I is incorrect.

Answer

To solve the problem, we need to analyze the function \( y = x + [x] \), where \([x]\) denotes the greatest integer function (also known as the floor function). We will evaluate the derivatives and verify the statements provided.

### Step 1: Understand the function
The function \( y = x + [x] \) consists of two parts:
- \( x \): a linear function.
- \([x]\): the greatest integer less than or equal to \( x \).

### Step 2: Find the derivative of \( y \)
To find the derivative \( \frac{dy}{dx} \), we need to differentiate both parts of the function.

1. The derivative of \( x \) is \( 1 \).
2. The derivative of \([x]\) is \( 0 \) for all \( x \) that is not an integer, and it does not exist at integer points.

Thus, we can express the derivative as:
\[ \frac{dy}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}([x]) = 1 + 0 = 1 \quad \text{for } x \text{ not an integer} \]

### Step 3: Analyze at integer points
At integer points, the derivative of \([x]\) does not exist. Therefore, the derivative \( \frac{dy}{dx} \) is not defined at these points. However, for all non-integer values of \( x \), we have:
\[ \frac{dy}{dx} = 1 \]

### Step 4: Evaluate the statements
- **Statement I**: \( \frac{dy}{dx} = 1 \) for all \( x \in \mathbb{R} \) is **false** because it does not hold true at integer values of \( x \).

- **Statement II**: The derivative of \([x]\) is \( 0 \) for non-integer \( x \) and does not exist at integer \( x \) is **true**.

### Conclusion
- Statement I is false.
- Statement II is true.