If positive numbers x, y, z satisfy `x+y+z=1` then
Statement 1: `[((1-2x)+(1-2y)+(1-2z))/(3)]ge((1-2x)(1-2y)(1-2z))^(1//3)`
because
Statement 2: For any three positive numbers a, b, c their `A.M. ge G.M.`

Options

A. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1
B. Statement - 1 is True, Statement - 2 is True, Statement -2 is NOT a correct explanation for Statement - 1
C. Statement - 1 is True, Statement - 2 is False
D. Statement - 1 is False, Statement - 2 is True

Answer

To solve the problem, we need to analyze the two statements given in the question regarding the positive numbers \( x, y, z \) that satisfy the equation \( x + y + z = 1 \).

### Step-by-Step Solution:

1. **Understand the Given Condition**:
We have three positive numbers \( x, y, z \) such that:
\[ x + y + z = 1 \]

2. **Rewrite Statement 1**:
The first statement is:
\[ \frac{(1 - 2x) + (1 - 2y) + (1 - 2z)}{3} \geq (1 - 2x)(1 - 2y)(1 - 2z)^{1/3} \]
We can simplify the left-hand side:
\[ \frac{(1 - 2x) + (1 - 2y) + (1 - 2z)}{3} = \frac{3 - 2(x + y + z)}{3} = \frac{3 - 2 \cdot 1}{3} = \frac{1}{3} \]
So, the left-hand side simplifies to \( \frac{1}{3} \).

3. **Evaluate the Right-Hand Side**:
Now, we need to evaluate the right-hand side:
\[ (1 - 2x)(1 - 2y)(1 - 2z)^{1/3} \]
To find this, we first calculate \( 1 - 2x, 1 - 2y, 1 - 2z \):
- Since \( x, y, z > 0 \), we have \( 1 - 2x > 0 \) if \( x < \frac{1}{2} \) (and similarly for \( y \) and \( z \)).
- The product \( (1 - 2x)(1 - 2y)(1 - 2z) \) is positive if all three terms are positive.

4. **Apply AM-GM Inequality**:
By the Arithmetic Mean-Geometric Mean (AM-GM) inequality, for any three positive numbers \( a, b, c \):
\[ \frac{a + b + c}{3} \geq (abc)^{1/3} \]
We can apply this inequality to \( A = 1 - 2x, B = 1 - 2y, C = 1 - 2z \):
\[ \frac{(1 - 2x) + (1 - 2y) + (1 - 2z)}{3} \geq (1 - 2x)(1 - 2y)(1 - 2z)^{1/3} \]
This confirms that the first statement is true.

5. **Conclusion**:
Since we have shown that the left-hand side is equal to \( \frac{1}{3} \) and the AM-GM inequality holds, we conclude that:
- Statement 1 is TRUE.
- Statement 2 correctly explains Statement 1, as it relies on the AM-GM inequality.

### Final Answer:
Both statements are true, and Statement 2 is a correct explanation for Statement 1.