The minimum value of `3^(x)+3^(1-x), x in R`

Options

A. `0`
B. `1/3`
C. `3`
D. `2sqrt(3)`

Answer

To find the minimum value of the expression \( 3^x + 3^{1-x} \) where \( x \) is a real number, we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ f(x) = 3^x + 3^{1-x} \]

2. **Apply the AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative real numbers \( a \) and \( b \):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
We can apply this to our expression by letting \( a = 3^x \) and \( b = 3^{1-x} \):
\[ \frac{3^x + 3^{1-x}}{2} \geq \sqrt{3^x \cdot 3^{1-x}} \]

3. **Simplify the Right Side**:
The right side simplifies as follows:
\[ \sqrt{3^x \cdot 3^{1-x}} = \sqrt{3^{x + (1-x)}} = \sqrt{3^1} = \sqrt{3} \]

4. **Multiply by 2**:
Now, we multiply both sides of the inequality by 2:
\[ 3^x + 3^{1-x} \geq 2\sqrt{3} \]

5. **Conclusion**:
The minimum value of \( 3^x + 3^{1-x} \) is \( 2\sqrt{3} \). This minimum value occurs when \( 3^x = 3^{1-x} \), which happens when \( x = \frac{1}{2} \).

### Final Answer:
The minimum value of \( 3^x + 3^{1-x} \) is \( 2\sqrt{3} \).