The minimum value of the expression `a+(1)/(a), a gt0` is :

Options

A. 0
B. 1
C. 2
D. 4

Answer

To find the minimum value of the expression \( a + \frac{1}{a} \) for \( a > 0 \), we can use the concept of the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-step Solution:

1. **Identify the Expression**: We have the expression \( a + \frac{1}{a} \) where \( a > 0 \).

2. **Apply AM-GM Inequality**: According to the AM-GM inequality, for any two positive numbers \( x \) and \( y \),
\[ \frac{x + y}{2} \geq \sqrt{xy} \]
Here, we can let \( x = a \) and \( y = \frac{1}{a} \).

3. **Set Up the Inequality**: Applying the AM-GM inequality, we have:
\[ \frac{a + \frac{1}{a}}{2} \geq \sqrt{a \cdot \frac{1}{a}} \]
Simplifying the right side gives:
\[ \sqrt{1} = 1 \]

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

5. **Conclusion**: The minimum value of \( a + \frac{1}{a} \) occurs when \( a + \frac{1}{a} = 2 \). This minimum value is achieved when \( a = 1 \) (since \( 1 + \frac{1}{1} = 2 \)).

Thus, the minimum value of the expression \( a + \frac{1}{a} \) for \( a > 0 \) is **2**.