`4 ^(sin ^(2)x ) + 4 ^(cos ^(2)x ) ge4` for all real x.

Answer

To solve the inequality \( 4^{\sin^2 x} + 4^{\cos^2 x} \geq 4 \) for all real \( x \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Identify the terms**: We have two terms, \( a = 4^{\sin^2 x} \) and \( b = 4^{\cos^2 x} \).

2. **Apply the AM-GM inequality**: According to the AM-GM inequality, for any non-negative numbers \( a \) and \( b \):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
Thus, we can write:
\[ \frac{4^{\sin^2 x} + 4^{\cos^2 x}}{2} \geq \sqrt{4^{\sin^2 x} \cdot 4^{\cos^2 x}} \]

3. **Simplify the right-hand side**: The right-hand side can be simplified using the property of exponents:
\[ \sqrt{4^{\sin^2 x} \cdot 4^{\cos^2 x}} = \sqrt{4^{\sin^2 x + \cos^2 x}} = \sqrt{4^1} = \sqrt{4} = 2 \]

4. **Multiply both sides by 2**: Now we multiply both sides of the inequality by 2:
\[ 4^{\sin^2 x} + 4^{\cos^2 x} \geq 2 \cdot 2 = 4 \]

5. **Conclusion**: Thus, we have shown that:
\[ 4^{\sin^2 x} + 4^{\cos^2 x} \geq 4 \]
for all real \( x \).

### Final Result:
The inequality \( 4^{\sin^2 x} + 4^{\cos^2 x} \geq 4 \) holds true for all real \( x \).