statement1:`e^pi` is bigger than `pi^e` statement 2:`f(x)=x^(1/x)` is an increasing function whenn `xe[e,oo)`

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 True
D. Statement-1 is False, Statement -2 is True

Answer

To solve the problem, we need to verify the two statements given:

1. **Statement 1**: \( e^{\pi} > \pi^{e} \)
2. **Statement 2**: The function \( f(x) = x^{\frac{1}{x}} \) is an increasing function for \( x \in [e, \infty) \).

### Step-by-step Solution:

#### Step 1: Analyze Statement 2
We start with the function:
\[ f(x) = x^{\frac{1}{x}} \]
To determine if \( f(x) \) is increasing or decreasing, we will differentiate it.

#### Step 2: Take the Natural Logarithm
Taking the natural logarithm of both sides gives:
\[ \ln(f(x)) = \frac{1}{x} \ln(x) \]

#### Step 3: Differentiate Using the Product Rule
Now, we differentiate \( \ln(f(x)) \) with respect to \( x \):
\[ \frac{d}{dx}(\ln(f(x))) = \frac{d}{dx}\left(\frac{\ln(x)}{x}\right) \]
Using the quotient rule:
\[ \frac{d}{dx}\left(\frac{\ln(x)}{x}\right) = \frac{x \cdot \frac{1}{x} - \ln(x) \cdot 1}{x^2} = \frac{1 - \ln(x)}{x^2} \]

#### Step 4: Apply the Chain Rule
Using the chain rule:
\[ \frac{f'(x)}{f(x)} = \frac{1 - \ln(x)}{x^2} \]
Thus,
\[ f'(x) = f(x) \cdot \frac{1 - \ln(x)}{x^2} \]

#### Step 5: Determine the Sign of \( f'(x) \)
To determine if \( f(x) \) is increasing or decreasing, we need to analyze the sign of \( f'(x) \):
- For \( x \geq e \), \( \ln(x) \geq 1 \), hence \( 1 - \ln(x) \leq 0 \).
- Therefore, \( f'(x) \leq 0 \) for \( x \geq e \).

This means \( f(x) \) is a decreasing function for \( x \in [e, \infty) \).

#### Conclusion for Statement 2
Since \( f(x) \) is decreasing for \( x \in [e, \infty) \), **Statement 2 is false**.

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#### Step 6: Analyze Statement 1
Now we need to verify:
\[ e^{\pi} > \pi^{e} \]
We can use the function \( f(x) = x^{\frac{1}{x}} \) again.

#### Step 7: Evaluate \( f(e) \) and \( f(\pi) \)
Since \( f(x) \) is decreasing for \( x \geq e \):
\[ f(e) > f(\pi) \]
This implies:
\[ e^{\frac{1}{e}} > \pi^{\frac{1}{\pi}} \]

#### Step 8: Raise Both Sides to the Power of \( e \pi \)
Raising both sides to the power of \( e \pi \):
\[ (e^{\frac{1}{e}})^{e \pi} > (\pi^{\frac{1}{\pi}})^{e \pi} \]
This simplifies to:
\[ e^{\pi} > \pi^{e} \]

#### Conclusion for Statement 1
Thus, **Statement 1 is true**.

### Final Conclusion
- **Statement 1**: True
- **Statement 2**: False

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