If `f''(x) le0 "for all" x in (a,b)` then f'(x)=0

Options

A. exactly once in (a,b)
B. at most once in (a,b)
C. at leat once
D. none of these

Explanation

To solve the problem, we need to analyze the implications of the given condition \( f''(x) \leq 0 \) for all \( x \) in the interval \( (a, b) \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
We are given that \( f''(x) \leq 0 \) for all \( x \) in the interval \( (a, b) \). This means that the second derivative of the function \( f(x) \) is non-positive throughout the interval.

**Hint**: Recall that if the second derivative is non-positive, the function is concave down.

2. **Implications for the First Derivative**:
Since \( f''(x) \leq 0 \), it implies that \( f'(x) \) is non-increasing in the interval \( (a, b) \). This means that as we move from \( a \) to \( b \), the slope of the function \( f(x) \) does not increase.

**Hint**: A non-increasing function can either be constant or decrease.

3. **Considering the First Derivative**:
Let's assume there are two distinct points \( x_1 \) and \( x_2 \) in \( (a, b) \) such that \( f'(x_1) = 0 \) and \( f'(x_2) = 0 \). Since \( f'(x) \) is non-increasing, if it is zero at two distinct points, it must be zero everywhere in between those points.

**Hint**: Think about the behavior of a non-increasing function that touches the x-axis.

4. **Applying the Mean Value Theorem**:
By the Mean Value Theorem (specifically Lagrange's Mean Value Theorem), if \( f'(x_1) = 0 \) and \( f'(x_2) = 0 \), then there exists at least one point \( c \) in \( (x_1, x_2) \) such that \( f''(c) = 0 \).

**Hint**: Remember that the Mean Value Theorem guarantees a point where the derivative of the function equals the average rate of change.

5. **Contradiction**:
However, this leads to a contradiction because we know that \( f''(x) \leq 0 \) for all \( x \) in \( (a, b) \). If \( f''(c) = 0 \) for some \( c \), it contradicts the fact that \( f''(x) \) is less than or equal to zero everywhere in that interval.

**Hint**: Think about what it means for a function to be non-positive and what happens if it equals zero at any point.

6. **Conclusion**:
Therefore, the only possibility is that \( f'(x) = 0 \) can occur at most once in the interval \( (a, b) \). If it were to occur more than once, we would reach a contradiction as shown.

**Hint**: Conclude by summarizing the implications of the behavior of the first and second derivatives.

### Final Answer:
The correct conclusion is that \( f'(x) = 0 \) holds at most once in the interval \( (a, b) \).