If `a and b` are two distinct real roots of the polynomial `f(x)` such that `a < b`, then there exists a real number c lying between `a and b`, such that

Options

A. `f(c) - 0`
B. `f'(c) = 0`
C. `f''(c) = 0`
D. none of these

Answer

To solve the problem, we will use Rolle's Theorem, which states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

### Step-by-Step Solution:

1. **Identify the roots**: We are given that `a` and `b` are two distinct real roots of the polynomial `f(x)`, with the condition that `a < b`. This means that `f(a) = 0` and `f(b) = 0`.

2. **Apply Rolle's Theorem**: Since `f(x)` is a polynomial, it is continuous and differentiable everywhere. Therefore, we can apply Rolle's Theorem on the interval [a, b].

3. **Verify the conditions of Rolle's Theorem**:
- The function `f(x)` is continuous on [a, b].
- The function `f(x)` is differentiable on (a, b).
- We have `f(a) = 0` and `f(b) = 0`.

4. **Conclude using Rolle's Theorem**: According to Rolle's Theorem, since the conditions are satisfied, there exists at least one real number `c` in the interval (a, b) such that:
\[ f'(c) = 0 \]

5. **Final Result**: Therefore, the required condition is that there exists a real number `c` lying between `a` and `b` such that:
\[ f'(c) = 0 \]