Let m and n be 2 positive integers, such that `m < n`. Which of the following compound inequalities must be true?

Options

A. `0 lt sqrt(mn) lt m`
B. `1 lt sqrt(mn) lt m`
C. `m lt sqrt(mn) lt n`
D. `sqrt(m) lt sqrt(mn) lt sqrt(n)`

Answer

To solve the problem, we need to analyze the inequalities given the condition that \( m \) and \( n \) are two positive integers such that \( m < n \). We will evaluate each of the options step by step.

### Step-by-Step Solution:

1. **Understanding the Condition**: We know that \( m \) and \( n \) are positive integers and \( m < n \). This means both \( m \) and \( n \) are greater than zero.

2. **Analyzing Option 1**:
- The inequality is \( 0 < \sqrt{mn} < m \).
- Since \( m \) and \( n \) are positive, \( \sqrt{mn} \) is also positive. Thus, \( 0 < \sqrt{mn} \) is true.
- However, since \( m < n \), we can say \( mn < n^2 \), which implies \( \sqrt{mn} < n \) but does not guarantee \( \sqrt{mn} < m \). Therefore, this option is **not necessarily true**.

3. **Analyzing Option 2**:
- The inequality is \( 1 < \sqrt{mn} < m \).
- While \( \sqrt{mn} \) is positive, we cannot guarantee that \( \sqrt{mn} \) is always greater than 1, especially if \( m \) is 1. Thus, this option is **not necessarily true**.

4. **Analyzing Option 3**:
- The inequality is \( m < \sqrt{mn} < n \).
- We know \( m < n \).
- If we square both sides of \( m < n \), we get \( m^2 < n^2 \).
- Multiplying the original inequality \( m < n \) by \( n \) gives us \( mn < n^2 \), hence \( \sqrt{mn} < n \) is true.
- Similarly, multiplying \( m < n \) by \( m \) gives \( m^2 < mn \), which implies \( m < \sqrt{mn} \) since both \( m \) and \( n \) are positive.
- Thus, this option is **true**.

5. **Analyzing Option 4**:
- The inequality is \( \sqrt{m} < \sqrt{mn} < \sqrt{n} \).
- Since \( m < n \), we know \( \sqrt{m} < \sqrt{n} \).
- However, we cannot guarantee that \( \sqrt{m} < \sqrt{mn} \) because \( \sqrt{mn} \) could be less than or equal to \( \sqrt{m} \) depending on the values of \( m \) and \( n \). Therefore, this option is **not necessarily true**.

### Conclusion:
The only compound inequality that must be true given the conditions is:

**Option 3: \( m < \sqrt{mn} < n \)**.