`f(x) = |x-1| + |x-2|, g(x) = x+(1)/(x), x gt 0` if `m_(1) = min (f(x))` and `m_(2) = min (g(x))`, then `(m_(1)+m_(2))/(m_(2)-m_(1))` is equal to_________.

Answer

To solve the problem, we need to find the minimum values of the functions \( f(x) = |x-1| + |x-2| \) and \( g(x) = x + \frac{1}{x} \) for \( x > 0 \). We will denote the minimum of \( f(x) \) as \( m_1 \) and the minimum of \( g(x) \) as \( m_2 \). Finally, we will calculate the expression \( \frac{m_1 + m_2}{m_2 - m_1} \).

### Step 1: Finding \( m_1 = \min f(x) \)

1. **Identify the critical points of \( f(x) \)**:
- The function \( f(x) \) involves absolute values, which change behavior at \( x = 1 \) and \( x = 2 \).
- We will consider three intervals: \( (-\infty, 1] \), \( (1, 2] \), and \( (2, \infty) \).

2. **Evaluate \( f(x) \) in each interval**:
- For \( x \leq 1 \):
\[ f(x) = |x-1| + |x-2| = (1-x) + (2-x) = 3 - 2x \]
- For \( 1 < x \leq 2 \):
\[ f(x) = |x-1| + |x-2| = (x-1) + (2-x) = 1 \]
- For \( x > 2 \):
\[ f(x) = |x-1| + |x-2| = (x-1) + (x-2) = 2x - 3 \]

3. **Determine the minimum value**:
- In the interval \( (-\infty, 1] \), as \( x \) approaches 1, \( f(x) \) approaches \( 1 \).
- In the interval \( (1, 2] \), \( f(x) = 1 \).
- In the interval \( (2, \infty) \), as \( x \) increases, \( f(x) \) increases.
- Therefore, the minimum value \( m_1 = 1 \).

### Step 2: Finding \( m_2 = \min g(x) \)

1. **Differentiate \( g(x) \)**:
\[ g(x) = x + \frac{1}{x} \]
\[ g'(x) = 1 - \frac{1}{x^2} \]

2. **Set the derivative to zero**:
\[ 1 - \frac{1}{x^2} = 0 \implies x^2 = 1 \implies x = 1 \quad (\text{since } x > 0) \]

3. **Evaluate \( g(x) \) at \( x = 1 \)**:
\[ g(1) = 1 + \frac{1}{1} = 2 \]

4. **Determine the behavior of \( g(x) \)**:
- For \( x < 1 \), \( g'(x) < 0 \) (decreasing).
- For \( x > 1 \), \( g'(x) > 0 \) (increasing).
- Thus, \( g(x) \) has a minimum at \( x = 1 \), so \( m_2 = 2 \).

### Step 3: Calculate \( \frac{m_1 + m_2}{m_2 - m_1} \)

1. **Substitute the values**:
\[ m_1 = 1, \quad m_2 = 2 \]
\[ \frac{m_1 + m_2}{m_2 - m_1} = \frac{1 + 2}{2 - 1} = \frac{3}{1} = 3 \]

### Final Answer:
\[ \frac{m_1 + m_2}{m_2 - m_1} = 3 \]