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Let f(x) be a non - constant polynomial ...

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

A

2

B

1

C

0

D

3

Text Solution

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The correct Answer is:
To solve the problem, we need to analyze the polynomial function \( f(x) \) given that it satisfies \( f(a) = f(b) = f(c) = 2 \) for distinct points \( a, b, c \). We aim to find the minimum number of roots of the equation \( f''(x) = 0 \) in the interval \( (a, c) \). ### Step-by-step Solution: 1. **Understanding the Polynomial**: Since \( f(x) \) is a non-constant polynomial and it takes the same value (2) at three distinct points \( a, b, c \), we can infer that the polynomial must have at least three roots when we consider the function \( g(x) = f(x) - 2 \). This function \( g(x) \) will have roots at \( a, b, c \). 2. **Applying Rolle's Theorem**: By Rolle's Theorem, since \( g(a) = g(b) = 0 \), there exists at least one point \( c_1 \) in the interval \( (a, b) \) such that \( g'(c_1) = 0 \). This means that \( f'(c_1) = 0 \). 3. **Finding More Critical Points**: Similarly, since \( g(b) = g(c) = 0 \), there exists at least one point \( c_2 \) in the interval \( (b, c) \) such that \( g'(c_2) = 0 \). This means that \( f'(c_2) = 0 \). 4. **Analyzing the Derivative**: Now we have two points \( c_1 \) and \( c_2 \) in the interval \( (a, c) \) where \( f'(x) = 0 \). Between these two points, we can apply Rolle's Theorem again. Since \( f'(c_1) = f'(c_2) = 0 \), there exists at least one point \( c_3 \) in the interval \( (c_1, c_2) \) such that \( f''(c_3) = 0 \). 5. **Conclusion**: Therefore, we have found that there is at least one root of \( f''(x) = 0 \) in the interval \( (a, c) \). Since we have established two critical points \( c_1 \) and \( c_2 \) and at least one point \( c_3 \) between them, 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 the interval \( (a, c) \) is **1**.
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