Statement-1 : If angles A, B, C of `DeltaABC` are acute,
then `cotA cotB cotCle(1)/(3sqrt(3)).`
Statement-2: If a, b, c are positive real numbers and `-ltmlt1`, then
`(a^(m)+b^(m)+c^(m))/(3)lt((a+b+c)/(3))^(m)`

Options

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

Explanation

To solve the problem, we need to analyze both statements and prove their validity step by step.

### Step 1: Analyze Statement 2
**Statement 2** states that if \( a, b, c \) are positive real numbers and \( m \) belongs to the interval \( (0, 1) \), then:
\[ \frac{a^m + b^m + c^m}{3} < \left( \frac{a + b + c}{3} \right)^m \]

**Proof:**
1. **Apply the Power Mean Inequality**: The Power Mean Inequality states that for \( m < n \):
\[ \left( \frac{x_1^n + x_2^n + ... + x_k^n}{k} \right)^{1/n} \geq \left( \frac{x_1^m + x_2^m + ... + x_k^m}{k} \right)^{1/m} \]
Here, we can take \( n = 1 \) and \( m \) as given, where \( 0 < m < 1 \).

2. **Apply the Inequality**: For \( a, b, c \):
\[ \left( \frac{a + b + c}{3} \right) \geq \left( \frac{a^m + b^m + c^m}{3} \right)^{1/m} \]

3. **Raise Both Sides to the Power of \( m \)**:
\[ \left( \frac{a + b + c}{3} \right)^m \geq \frac{a^m + b^m + c^m}{3} \]

4. **Conclusion**: Thus, Statement 2 is true.

### Step 2: Analyze Statement 1
**Statement 1** states that if angles \( A, B, C \) of triangle \( \Delta ABC \) are acute, then:
\[ \cot A \cdot \cot B \cdot \cot C \leq \frac{1}{3\sqrt{3}} \]

**Proof:**
1. **Use the Identity**: Since \( A + B + C = \pi \), we can use the identity:
\[ \tan(A + B + C) = \tan(\pi) = 0 \]
This gives us:
\[ \tan A + \tan B + \tan C = \tan A \tan B \tan C \]

2. **Apply AM-GM Inequality**: We apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality to \( \tan A, \tan B, \tan C \):
\[ \frac{\tan A + \tan B + \tan C}{3} \geq \sqrt[3]{\tan A \tan B \tan C} \]

3. **Substituting**: From the identity, we know \( \tan A + \tan B + \tan C = \tan A \tan B \tan C \). Thus:
\[ \frac{\tan A \tan B \tan C}{3} \geq \sqrt[3]{\tan A \tan B \tan C} \]

4. **Cubing Both Sides**: Cubing both sides gives:
\[ \frac{(\tan A \tan B \tan C)^3}{27} \geq \tan A \tan B \tan C \]

5. **Rearranging**: This leads to:
\[ (\tan A \tan B \tan C)^2 \geq 27 \]

6. **Taking Reciprocals**: Taking the reciprocal gives:
\[ \cot A \cot B \cot C \leq \frac{1}{3\sqrt{3}} \]

7. **Conclusion**: Thus, Statement 1 is true.

### Final Conclusion
Both statements are true, but Statement 2 does not serve as a direct explanation for Statement 1. Therefore, both statements are valid independently.