Let a differentiable function `f:RtoR` be such that for all x and y in R
`2|f(x)-f(y)|le|x-y|` and `f^(')(x)ge(1)/(2)`. So then the number of points of intersection of the graph `y=f(x)` with

Options

A. the line `y=x` is one.
B. the curve `y=-x^(3)` is one.
C. the curve `2y=|x|` is three.
D. the curve `y^(2)=-x` may be more than one.

Explanation

To solve the problem, we will analyze the given conditions step by step.

### Given:
1. A differentiable function \( f: \mathbb{R} \to \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \):
\[ 2 |f(x) - f(y)| \leq |x - y| \]
2. The derivative of the function satisfies:
\[ f'(x) \geq \frac{1}{2} \]

### Step 1: Analyze the first condition
From the first condition, we can rewrite it as:
\[ |f(x) - f(y)| \leq \frac{1}{2} |x - y| \]
This implies that the function \( f \) is Lipschitz continuous with a Lipschitz constant of \( \frac{1}{2} \). This means that the function cannot change too rapidly.

### Step 2: Analyze the second condition
The second condition states that the derivative \( f'(x) \) is always greater than or equal to \( \frac{1}{2} \). This means that the function is increasing and has a minimum slope of \( \frac{1}{2} \).

### Step 3: Combine the conditions
From the two conditions, we can conclude that:
- The function is Lipschitz continuous with a constant of \( \frac{1}{2} \).
- The function is increasing with a minimum slope of \( \frac{1}{2} \).

### Step 4: Determine the form of the function
Since \( f'(x) \geq \frac{1}{2} \), we can integrate this inequality:
\[ f(x) \geq \frac{1}{2}x + C \]
for some constant \( C \).

### Step 5: Find the points of intersection
Now, we need to find the number of points of intersection of the graph \( y = f(x) \) with various lines.

1. **For the line \( y = x \)**:
- The function \( f(x) \) is increasing and has a slope of at least \( \frac{1}{2} \), while the line \( y = x \) has a slope of \( 1 \).
- Therefore, there will be exactly one point of intersection.

2. **For the line \( y = -x^3 \)**:
- The function \( f(x) \) is increasing, while \( -x^3 \) is decreasing.
- Thus, there will be exactly one point of intersection.

3. **For the line \( y = |x| \)**:
- The function \( f(x) \) is increasing, while \( |x| \) has a V-shape and is not strictly increasing.
- Depending on the value of \( C \), there may be zero or one point of intersection, but not more than one.

4. **For the line \( y = -x \)**:
- Again, since \( f(x) \) is increasing and \( -x \) is decreasing, there may be one point of intersection.

### Conclusion
Based on the analysis, we conclude that:
- The number of points of intersection with \( y = x \) is **1**.
- The number of points of intersection with \( y = -x^3 \) is **1**.
- The number of points of intersection with \( y = |x| \) is **0 or 1**.
- The number of points of intersection with \( y = -x \) is **1**.

Thus, the correct options for the number of points of intersection are:
- **Option A**: 1 point with \( y = x \)
- **Option B**: 1 point with \( y = -x^3 \)
- **Option C**: Not more than 1 point with \( y = |x| \)
- **Option D**: May have more than one point with \( y = -x \)