If a, b, c `in` R, then which one of the following is true:

Options

A. `max(a,b)ltmin(a,b,c)`
B. `min(a,b)=(1)/(2){a+b-|a-b|}`
C. `min(a,b)ltmin(a,b,c)`
D. none of these

Explanation

To solve the problem, we need to analyze the given options one by one. The question states that if \( a, b, c \in \mathbb{R} \), we need to determine which of the following statements is true.

### Step 1: Analyze the first option
**Option 1:** \( \max(a, b) > \min(a, b, c) \)

Assume \( c < a \) and \( c < b \). In this case, \( \min(a, b, c) = c \) and \( \max(a, b) = a \) (assuming \( a > b \)).

Thus, we have:
\[ \max(a, b) = a \quad \text{and} \quad \min(a, b, c) = c \]
Since we assumed \( c < a \), it follows that:
\[ a > c \]
This means that the first option holds true under this assumption. However, we need to check if it holds for all cases.

If \( c \) is greater than both \( a \) and \( b \), then \( \min(a, b, c) = a \) or \( b \), which does not guarantee that \( \max(a, b) > \min(a, b, c) \). Hence, this option is not universally true.

### Step 2: Analyze the second option
**Option 2:** \( \min(a, b) = \frac{1}{2}(a + b - |a - b|) \)

Assume \( a < b \). Then:
\[ \min(a, b) = a \]
Now, calculate the right-hand side:
\[ \frac{1}{2}(a + b - |a - b|) = \frac{1}{2}(a + b - (b - a)) = \frac{1}{2}(2a) = a \]
This shows that the left-hand side equals the right-hand side when \( a < b \).

Now, assume \( b < a \):
\[ \min(a, b) = b \]
Calculating the right-hand side:
\[ \frac{1}{2}(a + b - |a - b|) = \frac{1}{2}(a + b - (a - b)) = \frac{1}{2}(2b) = b \]
Thus, in both cases, \( \min(a, b) \) equals the right-hand side. Therefore, this option is universally true.

### Step 3: Analyze the third option
**Option 3:** \( \min(a, b) < \min(a, b, c) \)

Assuming \( c < a \) and \( c < b \), we have:
\[ \min(a, b, c) = c \]
If we take \( b < a \), then:
\[ \min(a, b) = b \]
Since \( c < b \), it follows that:
\[ b < c \]
This contradicts the assumption that \( \min(a, b) < \min(a, b, c) \). Hence, this option is not universally true.

### Conclusion
After analyzing all options, we find that the second option is the only one that holds true for all cases.

**Final Answer:** The correct option is **Option 2**: \( \min(a, b) = \frac{1}{2}(a + b - |a - b|) \).