Statement-1: If f and g are differentiable at x=c, then min (f,g) is differentiable at x=c.
Statement-2: min (f,g) is differentiable at `x=c if f(c ) ne g(c )`

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To analyze the given statements, we need to evaluate the differentiability of the function \( \min(f, g) \) at a point \( x = c \) under the conditions provided.

### Step-by-Step Solution:

1. **Understanding the Functions**:
Let \( f(x) \) and \( g(x) \) be two differentiable functions at \( x = c \). We need to analyze the function \( h(x) = \min(f(x), g(x)) \).

2. **Differentiability of \( \min(f, g) \)**:
The function \( h(x) = \min(f(x), g(x)) \) will be differentiable at \( x = c \) if it is continuous and does not have a sharp corner at that point. A sharp corner occurs when \( f(c) = g(c) \).

3. **Case Analysis**:
- **Case 1**: If \( f(c) \neq g(c) \):
- In this case, \( h(x) \) will be equal to either \( f(x) \) or \( g(x) \) in a neighborhood around \( c \). Therefore, \( h(x) \) will inherit the differentiability from the function that is lesser at that point. Thus, \( h(x) \) is differentiable at \( x = c \).

- **Case 2**: If \( f(c) = g(c) \):
- Here, we need to check the behavior of \( h(x) \) around \( c \). Since both functions are equal at \( c \), we need to check the left-hand and right-hand derivatives. If \( f \) and \( g \) have different slopes at \( c \), \( h(x) \) will not be differentiable at \( c \) because it will create a corner point.

4. **Conclusion**:
- **Statement 1**: "If \( f \) and \( g \) are differentiable at \( x = c \), then \( \min(f, g) \) is differentiable at \( x = c \)" is **false** because it fails when \( f(c) = g(c) \) and they have different derivatives.
- **Statement 2**: "The function \( \min(f, g) \) is differentiable at \( x = c \) if \( f(c) \neq g(c) \)" is **true** because it holds under this condition.

### Final Answer:
- Statement 1 is false.
- Statement 2 is true.