Statement 1:` pi^(4) gt 4^(pi)`
because
Statement 2 : The function `y = x^(x)` is decreasing `forall x gt 1/e`.

Options

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

Answer

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

**Statement 1:** \( \pi^4 > 4^\pi \)
**Statement 2:** The function \( y = x^x \) is decreasing for all \( x > \frac{1}{e} \).

### Step 1: Analyze Statement 2

We start with the function \( y = x^x \). To determine if this function is decreasing for \( x > \frac{1}{e} \), we will differentiate it.

1. Take the natural logarithm of both sides:
\[ \ln y = x \ln x \]

2. Differentiate both sides with respect to \( x \):
\[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1 \]

3. Multiply both sides by \( y \) (which is \( x^x \)):
\[ \frac{dy}{dx} = x^x (\ln x + 1) \]

4. For \( y = x^x \) to be decreasing, we need \( \frac{dy}{dx} < 0 \):
\[ x^x (\ln x + 1) < 0 \]
Since \( x^x > 0 \) for \( x > 0 \), we need:
\[ \ln x + 1 < 0 \implies \ln x < -1 \implies x < \frac{1}{e} \]

Thus, the function \( y = x^x \) is decreasing for \( x < \frac{1}{e} \) and **not** for \( x > \frac{1}{e}**. Therefore, **Statement 2 is false**. ### Step 2: Analyze Statement 1 Now we will verify Statement 1: \( \pi^4 > 4^\pi \).

1. Calculate \( \pi^4 \):
\[ \pi \approx 3.14 \implies \pi^4 \approx (3.14)^4 \approx 97.66 \]

2. Calculate \( 4^\pi \):
\[ 4^\pi = (2^2)^\pi = 2^{2\pi} \approx 2^{6.28} \approx 70.73 \]

3. Compare the two values:
\[ 97.66 > 70.73 \]

Thus, \( \pi^4 > 4^\pi \) is true, meaning **Statement 1 is true**.

### Conclusion

- **Statement 1 is true.**
- **Statement 2 is false.**

### Final Answer:
- Statement 1 is true, Statement 2 is false.

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