`f: R rarr R, f(x)` is a differentiable function such that all its successive derivatives exist. `f'(x)` can be zero at discrete points only and `f(x)f''(x) le 0 AA x in R`
If` f'(x) ne 0`, then maximum number of real roots of `f''(x) = 0 ` is /are

Options

A. no real root
B. one
C. two
D. three

Answer

To solve the problem, we need to analyze the conditions given for the function \( f(x) \) and its derivatives. Let's break down the solution step by step.

### Step 1: Understanding the conditions
We are given that \( f'(x) \neq 0 \) for all \( x \) in \( R \) except at discrete points. This means that \( f'(x) \) can only be zero at isolated points, and it is non-zero elsewhere.

### Step 2: Analyzing \( f''(x) \)
We also know that \( f(x)f''(x) \leq 0 \) for all \( x \in R \). This implies that the product of \( f(x) \) and \( f''(x) \) is non-positive. This condition will help us understand the behavior of \( f(x) \) and \( f''(x) \).

### Step 3: Considering the implications of \( f'(x) \neq 0 \)
Since \( f'(x) \) is not equal to zero except at discrete points, it implies that \( f(x) \) is either strictly increasing or strictly decreasing in intervals between these discrete points.

### Step 4: Finding the roots of \( f''(x) = 0 \)
The equation \( f''(x) = 0 \) indicates points of inflection where the concavity of the function changes. The maximum number of real roots of \( f''(x) = 0 \) can be determined by the behavior of \( f'(x) \).

### Step 5: Analyzing the relationship between \( f'(x) \) and \( f''(x) \)
If \( f'(x) \) is non-zero, it means that \( f'(x) \) does not change sign except at the discrete points where it is zero. Therefore, \( f''(x) \) can only change sign at these points.

### Step 6: Conclusion on the maximum number of real roots
Since \( f'(x) \) can only be zero at discrete points and does not change sign elsewhere, \( f''(x) \) can change sign at most once between any two consecutive points where \( f'(x) = 0 \). Therefore, the maximum number of real roots of \( f''(x) = 0 \) is determined by the number of intervals created by these discrete points.

Given that \( f'(x) \) is non-zero in between these points, the maximum number of times \( f''(x) \) can equal zero is limited to one change of sign per interval. Thus, the maximum number of real roots of \( f''(x) = 0 \) is **1**.

### Final Answer
The maximum number of real roots of \( f''(x) = 0 \) is **1**.

---