Read the information given below and answer the questions that follow :
The following functions have been defined :
`la(x, y, z) ="min" (x+y, y+z)`
`"le" (x,y,z) ="max" (x-y, y-z)`
`ma (x,y,z) =(1//2) ["le"(x,y,z) +la (x,y,z)]`
Given that ` x gt y gt z gt 0`, which of the following is necessarily true?

Options

A. A) `la (x, y, z) lt "le" (x, y, z)`
B. B) `ma(x, y, z) lt "la" (x, y, z)`
C. C) `ma(x, y, z) lt "le" (x, y, z)`
D. D) Cannot be determined

Answer

To solve the problem, we need to analyze the given functions based on the conditions provided: \( x > y > z > 0 \).

### Step 1: Evaluate \( la(x, y, z) \)

The function \( la(x, y, z) \) is defined as:
\[ la(x, y, z) = \text{min}(x+y, y+z) \]

Given that \( x > y > z \), we can analyze the two expressions:
- \( x + y \)
- \( y + z \)

Since \( x > y \) and \( y > z \), it follows that:
\[ x + y > y + z \]

Thus, the minimum value will be:
\[ la(x, y, z) = y + z \]

### Step 2: Evaluate \( le(x, y, z) \)

The function \( le(x, y, z) \) is defined as:
\[ le(x, y, z) = \text{max}(x-y, y-z) \]

Now, we need to evaluate the two differences:
- \( x - y \)
- \( y - z \)

Since \( x > y \) and \( y > z \), it follows that:
\[ x - y > 0 \quad \text{and} \quad y - z > 0 \]

However, we cannot directly compare \( x - y \) and \( y - z \) without specific values. For example:
- If \( x = 8, y = 3, z = 2 \), then \( x - y = 5 \) and \( y - z = 1 \), so \( le(x, y, z) = 5 \).
- If \( x = 15, y = 14, z = 2 \), then \( x - y = 1 \) and \( y - z = 12 \), so \( le(x, y, z) = 12 \).

Thus, \( le(x, y, z) \) can vary and cannot be determined definitively.

### Step 3: Evaluate \( ma(x, y, z) \)

The function \( ma(x, y, z) \) is defined as:
\[ ma(x, y, z) = \frac{1}{2} \left[ le(x, y, z) + la(x, y, z) \right] \]

From the previous evaluations:
- We found \( la(x, y, z) = y + z \).
- We found that \( le(x, y, z) \) cannot be determined definitively.

Thus, \( ma(x, y, z) \) also cannot be determined definitively because it depends on \( le(x, y, z) \).

### Conclusion

Given the evaluations, we conclude that:

- \( la(x, y, z) = y + z \) is determined.
- \( le(x, y, z) \) is not determined.
- \( ma(x, y, z) \) is not determined.

Therefore, the correct answer is that we cannot definitively determine \( le(x, y, z) \) and consequently \( ma(x, y, z) \).

### Final Answer
The option that is necessarily true is that we cannot determine the value of \( le(x, y, z) \).

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