If `f(x)` is a differentiable function wherever it is continuous and `f'(c_1)=f'(c_2)=0` `f''(c_1).f''(c_2)<0, f(c_1)=5,f(c_2)=0` and `(c_1ltc_2)` . If `f(x) ` is continuous in `[c_1,c_2]` and `f''(c_1)-f''(c_2)>0` then minimum number of roots of `f'(x)=0` in `[c_1-1,c_2+1]` is

Options

A. a. 1
B. b. 2
C. c. 3
D. d. 4

Explanation

To solve the problem step by step, we need to analyze the given information about the function \( f(x) \) and its derivatives.

### Step 1: Understand the given conditions
We have:
- \( f'(c_1) = f'(c_2) = 0 \)
- \( f''(c_1) \cdot f''(c_2) < 0 \)
- \( f(c_1) = 5 \) and \( f(c_2) = 0 \)
- \( c_1 < c_2 \)
- \( f''(c_1) - f''(c_2) > 0 \)

### Step 2: Analyze the second derivative conditions
From \( f''(c_1) \cdot f''(c_2) < 0 \), we can conclude that one of the second derivatives is positive and the other is negative. Given that \( f''(c_1) - f''(c_2) > 0 \), it implies:
- \( f''(c_1) > f''(c_2) \)
- Therefore, \( f''(c_1) > 0 \) and \( f''(c_2) < 0 \)

### Step 3: Determine the nature of critical points
- Since \( f''(c_1) > 0 \), \( c_1 \) is a local minimum.
- Since \( f''(c_2) < 0 \), \( c_2 \) is a local maximum.

### Step 4: Analyze the behavior of \( f'(x) \)
Since \( f'(c_1) = 0 \) and \( f'(c_2) = 0 \), and knowing the nature of these points:
- As we move from \( c_1 \) to \( c_2 \), \( f'(x) \) must increase to reach the maximum at \( c_2 \) and then decrease after \( c_2 \).

### Step 5: Apply the Intermediate Value Theorem
Given that \( f'(x) \) changes from negative to positive as it approaches \( c_1 \) (local minimum) and then back to negative as it approaches \( c_2 \) (local maximum), we can conclude that there must be at least one root of \( f'(x) = 0 \) in the intervals:
- \( (c_1 - 1, c_1) \)
- \( (c_1, c_2) \)
- \( (c_2, c_2 + 1) \)

### Step 6: Count the roots
1. **In the interval \( (c_1 - 1, c_1) \)**: Since \( f'(c_1) = 0 \) and \( f'(x) \) must increase to reach this point, there is at least one root.
2. **In the interval \( (c_1, c_2) \)**: Since \( f'(c_1) = 0 \) (local minimum) and \( f'(c_2) = 0 \) (local maximum), there is at least one root.
3. **In the interval \( (c_2, c_2 + 1) \)**: Since \( f'(c_2) = 0 \) and \( f'(x) \) must decrease after this point, there is at least one root.

### Conclusion
Thus, the minimum number of roots of \( f'(x) = 0 \) in the interval \( [c_1 - 1, c_2 + 1] \) is **3**.