A continuous and differentiable function `y=f(x)` is such that its graph cuts line `y=m x+c` at `n` distinct points. Then the minimum number of points at which `f^('')(x)=0` is/are

Options

A. `n-1`
B. `n-3`
C. `n-2`
D. cannot say

Answer

To solve the problem step by step, we need to analyze the given information about the function \( y = f(x) \) and the line \( y = mx + c \).

### Step 1: Understand the problem
We have a continuous and differentiable function \( y = f(x) \) that intersects the line \( y = mx + c \) at \( n \) distinct points. We need to determine the minimum number of points where the second derivative \( f''(x) = 0 \).

### Step 2: Identify the implications of the intersections
Since the function \( f(x) \) intersects the line \( y = mx + c \) at \( n \) distinct points, we can denote these points as \( x_1, x_2, \ldots, x_n \). At each of these points, we have:
\[ f(x_i) = mx_i + c \quad \text{for } i = 1, 2, \ldots, n \]

### Step 3: Apply the Mean Value Theorem
According to the Mean Value Theorem (MVT), if \( f(x) \) is continuous on the closed interval \([x_i, x_{i+1}]\) and differentiable on the open interval \((x_i, x_{i+1})\), there exists at least one point \( c_i \) in \((x_i, x_{i+1})\) such that:
\[ f'(c_i) = \frac{f(x_{i+1}) - f(x_i)}{x_{i+1} - x_i} \]
Since \( f(x_{i+1}) = mx_{i+1} + c \) and \( f(x_i) = mx_i + c \), we can simplify this to:
\[ f'(c_i) = m \]
for each interval \( [x_i, x_{i+1}] \).

### Step 4: Count the intervals
With \( n \) distinct points, we have \( n - 1 \) intervals:
1. \( [x_1, x_2] \)
2. \( [x_2, x_3] \)
3. ...
4. \( [x_{n-1}, x_n] \)

### Step 5: Apply Rolle's Theorem
Since \( f'(c_i) = m \) for each of the \( n - 1 \) intervals, we can apply Rolle's Theorem to each interval. This theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point \( d_i \) in \( (x_i, x_{i+1}) \) such that:
\[ f''(d_i) = 0 \]
Thus, for \( n - 1 \) intervals, we will have at least \( n - 2 \) points where \( f''(x) = 0 \).

### Conclusion
Therefore, the minimum number of points at which \( f''(x) = 0 \) is:
\[ \boxed{n - 2} \]