Statement-1: The equation `(pi^(e))/(x-e)+(e^(pi))/(x-pi)+(pi^(pi)+e^(e))/(x-pi-e) = 0` has real roots.
Statement-2: If f(x) is a polynomial and a, b are two real numbers such that `f(a) f(b) lt 0`, then f(x) = 0 has an odd number of real roots between a and b.

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.

Explanation

To solve the problem, we need to analyze both statements provided.

### Step 1: Analyze Statement 2
Statement 2 states that if \( f(x) \) is a polynomial and \( a, b \) are two real numbers such that \( f(a) \cdot f(b) < 0 \), then \( f(x) = 0 \) has an odd number of real roots between \( a \) and \( b \).

**Justification:**
This is a standard result in calculus known as the Intermediate Value Theorem. If \( f(a) \) is positive and \( f(b) \) is negative (or vice versa), the polynomial must cross the x-axis at least once between \( a \) and \( b \). Since polynomials are continuous functions, they can cross the x-axis an odd number of times (1, 3, 5, etc.) in that interval. Thus, Statement 2 is true.

### Step 2: Analyze Statement 1
Statement 1 states that the equation

\[ \frac{\pi^e}{x-e} + \frac{e^\pi}{x-\pi} + \frac{\pi^\pi + e^e}{x-\pi-e} = 0 \]

has real roots.

**Let’s define the function:**
Let

\[ f(x) = \frac{\pi^e}{x-e} + \frac{e^\pi}{x-\pi} + \frac{\pi^\pi + e^e}{x-\pi-e} \]

We need to determine if this function has real roots.

### Step 3: Evaluate \( f(e) \), \( f(\pi) \), and \( f(\pi + e) \)

1. **Calculate \( f(e) \):**
\[ f(e) = \frac{\pi^e}{e-e} + \frac{e^\pi}{e-\pi} + \frac{\pi^\pi + e^e}{e-\pi-e} \]
The first term is undefined (division by zero), but we can analyze the limit as \( x \) approaches \( e \) from the right. As \( x \to e^+ \), \( f(e) \to +\infty \).

2. **Calculate \( f(\pi) \):**
\[ f(\pi) = \frac{\pi^e}{\pi-e} + \frac{e^\pi}{\pi-\pi} + \frac{\pi^\pi + e^e}{\pi-\pi-e} \]
The second term is undefined (division by zero), but we can analyze the limit as \( x \) approaches \( \pi \) from the left. As \( x \to \pi^- \), \( f(\pi) \to -\infty \).

3. **Calculate \( f(\pi + e) \):**
\[ f(\pi + e) = \frac{\pi^e}{\pi + e - e} + \frac{e^\pi}{\pi + e - \pi} + \frac{\pi^\pi + e^e}{\pi + e - \pi - e} \]
This is defined and can be calculated, but we can also analyze the behavior of \( f(x) \) in the intervals.

### Step 4: Conclusion
From the evaluations:
- \( f(e) \to +\infty \)
- \( f(\pi) \to -\infty \)
- \( f(\pi + e) \) is positive.

Since \( f(e) > 0 \) and \( f(\pi) < 0 \), by the Intermediate Value Theorem (which is supported by Statement 2), there is at least one real root in the interval \( (e, \pi) \).

### Final Answer
Both Statement 1 and Statement 2 are true. Statement 2 supports the conclusion for Statement 1.

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