Let a and b be two distinct roots of a polynomial equation f (x) = 0, Then there exists at least one root lying between a and b of polynomial equation.

Options

A. a) f(x) =0
B. b) f'(x) 0
C. c) f''(x) =0
D. d) None of these

Answer

To solve the problem, we will use the Intermediate Value Theorem, which is a fundamental theorem in calculus. The theorem states that if a function is continuous on a closed interval [a, b] and takes on different values at the endpoints, then it must take on every value between those endpoints at least once.

### Step-by-Step Solution:

1. **Identify the Function and Roots**:
Let \( f(x) \) be a polynomial function such that \( f(a) = 0 \) and \( f(b) = 0 \), where \( a \) and \( b \) are two distinct roots of the polynomial.

2. **Check Continuity**:
Since \( f(x) \) is a polynomial, it is continuous everywhere on the real line, including the interval \([a, b]\).

3. **Evaluate the Function at the Endpoints**:
We know that:
- \( f(a) = 0 \)
- \( f(b) = 0 \)

4. **Apply the Intermediate Value Theorem**:
According to the Intermediate Value Theorem, since \( f(x) \) is continuous on \([a, b]\) and \( f(a) \) and \( f(b) \) are both equal to 0, there must be at least one point \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \).

5. **Conclusion**:
Therefore, we conclude that there exists at least one root of the polynomial \( f(x) = 0 \) in the interval \((a, b)\).