If a, b, c, d are are positive real numbers such that a+b+c+d=2, then M=(a+b)(c+d) satisfies the relation is:

Options

A. `0leMle1`
B. `1 le M le 2`
C. `2 le M le 3`
D. `3leMle4`

Answer

To solve the problem, we need to find the maximum value of \( M = (a+b)(c+d) \) given that \( a + b + c + d = 2 \) and \( a, b, c, d \) are positive real numbers.

### Step-by-step Solution:

1. **Start with the given condition**:
\[ a + b + c + d = 2 \]
We can rearrange this to express \( c + d \) in terms of \( a + b \):
\[ c + d = 2 - (a + b) \]

2. **Substitute into the expression for \( M \)**:
\[ M = (a + b)(c + d) = (a + b)(2 - (a + b)) \]
Let \( x = a + b \). Then we can rewrite \( M \) as:
\[ M = x(2 - x) = 2x - x^2 \]

3. **Find the maximum value of \( M \)**:
The expression \( M = 2x - x^2 \) is a quadratic equation that opens downwards (since the coefficient of \( x^2 \) is negative). The maximum value occurs at the vertex of the parabola. The vertex \( x \) can be found using the formula:
\[ x = -\frac{b}{2a} = -\frac{2}{-2} = 1 \]

4. **Calculate \( M \) at \( x = 1 \)**:
Substitute \( x = 1 \) back into the equation for \( M \):
\[ M = 2(1) - (1)^2 = 2 - 1 = 1 \]

5. **Check the conditions**:
Since \( a + b = 1 \) and \( c + d = 1 \) (because \( c + d = 2 - (a + b) \)), we can choose \( a = b = 0.5 \) and \( c = d = 0.5 \) to satisfy the condition that all variables are positive real numbers.

6. **Conclusion**:
The maximum value of \( M \) is \( 1 \), and it can be achieved when \( a = b = c = d = 0.5 \).

### Final Result:
\[ M \leq 1 \]
The maximum value of \( M \) is \( 1 \).