`A(x, y, z)=" min. "(x+y, y+z, z+x)`
`B(x, y, z)="max "(x-y, y-z, z-x)`
`C(x, y, z)=" max"(A(x, y, z), B(x, y, z))`
`D(x, y, z)=" min "(A(x, y, z), B(x, y, z))`
The value of `C(1, 2, 3)` is :

Options

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

Answer

To solve the problem step by step, we need to evaluate the functions A, B, C, and D as defined in the question. We will specifically focus on calculating \( C(1, 2, 3) \).

### Step 1: Calculate \( A(1, 2, 3) \)

The function \( A(x, y, z) \) is defined as:
\[ A(x, y, z) = \min(x+y, y+z, z+x) \]

Substituting \( x = 1 \), \( y = 2 \), and \( z = 3 \):
- \( x + y = 1 + 2 = 3 \)
- \( y + z = 2 + 3 = 5 \)
- \( z + x = 3 + 1 = 4 \)

Now, we find the minimum of these values:
\[ A(1, 2, 3) = \min(3, 5, 4) = 3 \]

### Step 2: Calculate \( B(1, 2, 3) \)

The function \( B(x, y, z) \) is defined as:
\[ B(x, y, z) = \max(x-y, y-z, z-x) \]

Substituting \( x = 1 \), \( y = 2 \), and \( z = 3 \):
- \( x - y = 1 - 2 = -1 \)
- \( y - z = 2 - 3 = -1 \)
- \( z - x = 3 - 1 = 2 \)

Now, we find the maximum of these values:
\[ B(1, 2, 3) = \max(-1, -1, 2) = 2 \]

### Step 3: Calculate \( C(1, 2, 3) \)

The function \( C(x, y, z) \) is defined as:
\[ C(x, y, z) = \max(A(x, y, z), B(x, y, z)) \]

Using the values we calculated:
- \( A(1, 2, 3) = 3 \)
- \( B(1, 2, 3) = 2 \)

Now, we find the maximum:
\[ C(1, 2, 3) = \max(3, 2) = 3 \]

### Final Answer

The value of \( C(1, 2, 3) \) is \( 3 \).

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