The minimum value of `4^(x)+4^(1-x),x in RR` is

Options

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

Explanation

To find the minimum value of the expression \(4^x + 4^{1-x}\), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ y = 4^x + 4^{1-x} \]

2. **Apply AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative numbers \(a\) and \(b\):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
Here, let \(a = 4^x\) and \(b = 4^{1-x}\).

3. **Calculate the Product**:
The product \(ab\) is:
\[ ab = 4^x \cdot 4^{1-x} = 4^{x + (1-x)} = 4^1 = 4 \]

4. **Apply AM-GM**:
Now, applying the AM-GM inequality:
\[ \frac{4^x + 4^{1-x}}{2} \geq \sqrt{4} \]
This simplifies to:
\[ \frac{4^x + 4^{1-x}}{2} \geq 2 \]

5. **Multiply by 2**:
Multiplying both sides by 2 gives:
\[ 4^x + 4^{1-x} \geq 4 \]

6. **Conclusion**:
Thus, the minimum value of \(4^x + 4^{1-x}\) is \(4\).

### Final Answer:
The minimum value of \(4^x + 4^{1-x}\) is **4**.

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