Let f(x) denotes the number of zeroes in f'(x). If f(m)-f(n)=3, the value of
`((m-n)_(max)-(m-n)_(min))/(2)` is ........... .

Answer

To solve the problem, we need to analyze the function \( f(x) \) which denotes the number of zeroes in its derivative \( f'(x) \). We are given that \( f(m) - f(n) = 3 \). We need to find the value of

\[ \frac{(m-n)_{\text{max}} - (m-n)_{\text{min}}}{2}. \]

### Step 1: Understand the relationship between \( f(m) \) and \( f(n) \)

Since \( f(m) - f(n) = 3 \), it implies that the difference in the number of zeroes of \( f'(x) \) between \( m \) and \( n \) is 3. This means that there are 3 more zeroes in \( f'(x) \) between \( m \) and \( n \).

### Step 2: Determine the maximum value of \( m - n \)

The maximum difference \( (m - n)_{\text{max}} \) occurs when we have the maximum number of zeroes in \( f'(x) \) between \( m \) and \( n \). Since \( f(x) \) can have at most \( f(m) \) zeroes, and since \( f(m) - f(n) = 3 \), we can say that:

\[ (m - n)_{\text{max}} = 3. \]

### Step 3: Determine the minimum value of \( m - n \)

The minimum difference \( (m - n)_{\text{min}} \) occurs when there is just enough space to accommodate the 3 zeroes. The minimum number of zeroes that can be present while still having a difference of 3 is when the zeroes are as close as possible. The minimum gap that can accommodate 3 zeroes is 2. Therefore:

\[ (m - n)_{\text{min}} = 1. \]

### Step 4: Calculate the required expression

Now we can substitute the values we found into the expression:

\[ \frac{(m-n)_{\text{max}} - (m-n)_{\text{min}}}{2} = \frac{3 - 1}{2} = \frac{2}{2} = 1. \]

### Final Answer

Thus, the value of

\[ \frac{(m-n)_{\text{max}} - (m-n)_{\text{min}}}{2} = 1. \]