Which one of the following is correct?

Options

A. `x+(1)/(x) le-2`
B. `x+(1)/(x)=0`
C. `x+(1)/(x)ge2`
D. both (a) and (c)

Answer

To determine which option is correct, we will analyze the relationship between the arithmetic mean (AM) and the geometric mean (GM) of two numbers. The specific numbers we will consider are \( x \) and \( \frac{1}{x} \).

### Step-by-Step Solution:

1. **Define the Arithmetic Mean (AM)**:
The arithmetic mean of two numbers \( a \) and \( b \) is given by:
\[ AM = \frac{a + b}{2} \]
In our case, let \( a = x \) and \( b = \frac{1}{x} \). Therefore, the arithmetic mean becomes:
\[ AM = \frac{x + \frac{1}{x}}{2} \]

2. **Define the Geometric Mean (GM)**:
The geometric mean of two numbers \( a \) and \( b \) is given by:
\[ GM = \sqrt{a \cdot b} \]
For our numbers, the geometric mean is:
\[ GM = \sqrt{x \cdot \frac{1}{x}} = \sqrt{1} = 1 \]

3. **Apply the AM-GM Inequality**:
According to the AM-GM inequality, we have:
\[ AM \geq GM \]
Substituting the expressions we found:
\[ \frac{x + \frac{1}{x}}{2} \geq 1 \]

4. **Multiply Both Sides by 2**:
To eliminate the fraction, multiply both sides by 2:
\[ x + \frac{1}{x} \geq 2 \]

5. **Conclusion**:
The inequality \( x + \frac{1}{x} \geq 2 \) holds true for all positive values of \( x \). This confirms that the arithmetic mean of \( x \) and \( \frac{1}{x} \) is always greater than or equal to their geometric mean.

Thus, the correct option is **Option 3**.