Which one of the following is correct?

Options

A. `x+1/xle-2`
B. x+1/x=0
C. `x+1//xge2`
D. both(a) and©

Answer

To solve the question, we need to evaluate the given expressions for \( x = 1 \) and determine which statements are correct. The expressions we need to check are:

1. \( x + \frac{1}{x} \leq -2 \)
2. \( x + \frac{1}{x} = 0 \)
3. \( x + \frac{1}{x} \geq 2 \)

### Step-by-Step Solution:

**Step 1: Evaluate the first expression \( x + \frac{1}{x} \leq -2 \)**

- Substitute \( x = 1 \):
\[ 1 + \frac{1}{1} = 1 + 1 = 2 \]
- Now check if \( 2 \leq -2 \):
\[ 2 \leq -2 \quad \text{(This is false)} \]

**Conclusion for Step 1**: The first statement is incorrect.

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**Step 2: Evaluate the second expression \( x + \frac{1}{x} = 0 \)**

- Substitute \( x = 1 \):
\[ 1 + \frac{1}{1} = 1 + 1 = 2 \]
- Now check if \( 2 = 0 \):
\[ 2 = 0 \quad \text{(This is false)} \]

**Conclusion for Step 2**: The second statement is incorrect.

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**Step 3: Evaluate the third expression \( x + \frac{1}{x} \geq 2 \)**

- Substitute \( x = 1 \):
\[ 1 + \frac{1}{1} = 1 + 1 = 2 \]
- Now check if \( 2 \geq 2 \):
\[ 2 \geq 2 \quad \text{(This is true)} \]

**Conclusion for Step 3**: The third statement is correct.

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### Final Conclusion:
The only correct statement is option C: \( x + \frac{1}{x} \geq 2 \).

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