If `p+1/P=q` then for p gt 0 :

Options

A. q = 0
B. `-2ltqlt2`
C. `q ge 2`
D. none of a,b,c

Answer

To solve the equation \( p + \frac{1}{p} = q \) for \( p > 0 \), we will use the concept of the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression given in the question:
\[ p + \frac{1}{p} = q \]
Here, \( p \) is a positive number.

2. **Applying the AM-GM Inequality**:
According to the AM-GM inequality:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
for any non-negative \( a \) and \( b \). In our case, let \( a = p \) and \( b = \frac{1}{p} \).

3. **Calculating the Arithmetic Mean**:
The arithmetic mean (AM) of \( p \) and \( \frac{1}{p} \) is:
\[ \frac{p + \frac{1}{p}}{2} \]

4. **Calculating the Geometric Mean**:
The geometric mean (GM) of \( p \) and \( \frac{1}{p} \) is:
\[ \sqrt{p \cdot \frac{1}{p}} = \sqrt{1} = 1 \]

5. **Setting Up the Inequality**:
By the AM-GM inequality:
\[ \frac{p + \frac{1}{p}}{2} \geq 1 \]
Multiplying both sides by 2 gives:
\[ p + \frac{1}{p} \geq 2 \]

6. **Substituting Back to Find \( q \)**:
Since \( p + \frac{1}{p} = q \), we can substitute this into our inequality:
\[ q \geq 2 \]

7. **Conclusion**:
Therefore, for \( p > 0 \), we conclude that:
\[ q \geq 2 \]

### Answer:
The correct option is \( q \geq 2 \).