Let `f(x)` be a non - constant polynomial such that `f(a)=f(b)=f(c)=2.` Then the minimum number of roots of the equation `f''(x)=0" in "x in (a, c)` is/are

Options

A. 2
B. 1
C. 0
D. 3

Answer

To solve the problem, we need to determine the minimum number of roots of the equation \( f''(x) = 0 \) in the interval \( (a, c) \), given that \( f(a) = f(b) = f(c) = 2 \) for a non-constant polynomial \( f(x) \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that \( f(x) \) is a non-constant polynomial and it takes the same value (2) at three distinct points \( a, b, c \). This means that the graph of \( f(x) \) intersects the horizontal line \( y = 2 \) at three points.

2. **Applying Rolle's Theorem**:
Since \( f(a) = f(b) \) and \( f(b) = f(c) \), we can apply Rolle's Theorem:
- Between \( a \) and \( b \): There exists at least one point \( x_1 \in (a, b) \) such that \( f'(x_1) = 0 \).
- Between \( b \) and \( c \): There exists at least one point \( x_2 \in (b, c) \) such that \( f'(x_2) = 0 \).

3. **Finding Roots of the First Derivative**:
From the application of Rolle's Theorem, we have established that there are at least two points where \( f'(x) = 0 \):
- One root in \( (a, b) \)
- Another root in \( (b, c) \)

4. **Analyzing the Second Derivative**:
The points where \( f'(x) = 0 \) indicate local maxima or minima. To find the points where \( f''(x) = 0 \), we need to consider the behavior of \( f'(x) \):
- If \( f'(x) \) has two roots, it must change direction at least once between these roots, which implies that \( f''(x) \) must be zero at least once between these two points.

5. **Conclusion**:
Therefore, since we have two roots of \( f'(x) \) in the interval \( (a, c) \), and \( f''(x) \) must be zero at least once between these roots, we conclude that the minimum number of roots of the equation \( f''(x) = 0 \) in the interval \( (a, c) \) is **1**.

### Final Answer:
The minimum number of roots of the equation \( f''(x) = 0 \) in \( (a, c) \) is **1**.