`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))`
What is the value of `A(2, 3, 4)+B(2, 3, 4)`?

Options

A. 13
B. 4
C. 7
D. none of these

Answer

To solve the problem, we need to evaluate the functions A and B for the inputs \(x = 2\), \(y = 3\), and \(z = 4\), and then find the sum \(A(2, 3, 4) + B(2, 3, 4)\).

### Step-by-step solution:

1. **Calculate \(A(2, 3, 4)\)**:
\[ A(x, y, z) = \text{min}(x+y, y+z, z+x) \]
Substituting \(x = 2\), \(y = 3\), and \(z = 4\):
\[ A(2, 3, 4) = \text{min}(2+3, 3+4, 4+2) \]
Calculate each term:
- \(2 + 3 = 5\)
- \(3 + 4 = 7\)
- \(4 + 2 = 6\)

Now find the minimum:
\[ A(2, 3, 4) = \text{min}(5, 7, 6) = 5 \]

2. **Calculate \(B(2, 3, 4)\)**:
\[ B(x, y, z) = \text{max}(x-y, y-z, z-x) \]
Substituting \(x = 2\), \(y = 3\), and \(z = 4\):
\[ B(2, 3, 4) = \text{max}(2-3, 3-4, 4-2) \]
Calculate each term:
- \(2 - 3 = -1\)
- \(3 - 4 = -1\)
- \(4 - 2 = 2\)

Now find the maximum:
\[ B(2, 3, 4) = \text{max}(-1, -1, 2) = 2 \]

3. **Calculate \(A(2, 3, 4) + B(2, 3, 4)\)**:
\[ A(2, 3, 4) + B(2, 3, 4) = 5 + 2 = 7 \]

### Final Answer:
The value of \(A(2, 3, 4) + B(2, 3, 4)\) is \(7\).

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