Statement-1 `e^(pi) gt pi^( e)`
Statement -2 The function `x^(1//x)( x gt 0)` is strictly decreasing in `[e ,oo)`

Options

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

Explanation

To solve the problem, we need to analyze both statements given:

### Statement 1: \( e^{\pi} > \pi^{e} \)

### Statement 2: The function \( f(x) = x^{\frac{1}{x}} \) is strictly decreasing in the interval \( [e, \infty) \).

### Step-by-step Solution:

**Step 1: Analyze Statement 2**

We start by analyzing the function \( f(x) = x^{\frac{1}{x}} \).

1. **Take the natural logarithm of both sides:**
\[ \ln(f(x)) = \frac{1}{x} \ln(x) \]

2. **Differentiate \( f(x) \):**
Using the chain rule, we differentiate \( \ln(f(x)) \):
\[ \frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)} \]
Therefore,
\[ f'(x) = f(x) \cdot \frac{d}{dx} \left(\frac{\ln(x)}{x}\right) \]

3. **Differentiate \( \frac{\ln(x)}{x} \):**
Using the quotient rule:
\[ \frac{d}{dx} \left(\frac{\ln(x)}{x}\right) = \frac{1 \cdot x - \ln(x) \cdot 1}{x^2} = \frac{1 - \ln(x)}{x^2} \]

4. **Substituting back:**
\[ f'(x) = f(x) \cdot \frac{1 - \ln(x)}{x^2} \]

5. **Determine the sign of \( f'(x) \):**
- \( f(x) > 0 \) for \( x > 0 \).
- The critical point occurs when \( 1 - \ln(x) = 0 \) or \( \ln(x) = 1 \) which gives \( x = e \).

6. **Test intervals around \( x = e \):**
- For \( x < e \): \( 1 - \ln(x) > 0 \) implies \( f'(x) > 0 \) (function is increasing).
- For \( x > e \): \( 1 - \ln(x) < 0 \) implies \( f'(x) < 0 \) (function is decreasing).

Thus, \( f(x) \) is strictly decreasing in the interval \( [e, \infty) \).

### Conclusion for Statement 2:
Statement 2 is true.

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**Step 2: Analyze Statement 1**

Now we use the information from Statement 2 to analyze Statement 1.

1. **Evaluate \( f(e) \) and \( f(\pi) \):**
- Since \( f(x) \) is strictly decreasing in \( [e, \infty) \) and \( \pi > e \):
\[ f(\pi) < f(e) \]

2. **Calculate \( f(e) \) and \( f(\pi) \):**
- \( f(e) = e^{\frac{1}{e}} \)
- \( f(\pi) = \pi^{\frac{1}{\pi}} \)

3. **Using the property of the function:**
Since \( f(\pi) < f(e) \), we have:
\[ \pi^{\frac{1}{\pi}} < e^{\frac{1}{e}} \]

4. **Exponentiating both sides:**
Raising both sides to the power of \( e \) and \( \pi \) respectively gives:
\[ e^{\pi} > \pi^{e} \]

### Conclusion for Statement 1:
Statement 1 is true.

### Final Conclusion:
Both statements are true, and Statement 2 correctly explains Statement 1.

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