Consider a differentiable `f:R to R` for which `f(1)=2 and f(x+y)=2^(x)f(y)+4^(y)f(x) AA x , y in R.`
The minimum value of `f(x)` is

Options

A. `1`
B. `-(1)/(2)`
C. `-(1)/(4)`
D. none of these

Explanation

To solve the problem, we need to analyze the functional equation given and derive the function \( f(x) \).

### Step 1: Write down the functional equation
The functional equation given is:
\[ f(x+y) = 2^x f(y) + 4^y f(x) \quad \text{for all } x, y \in \mathbb{R} \]
We also know that \( f(1) = 2 \).

### Step 2: Substitute specific values
Let's first substitute \( y = 0 \) into the equation:
\[ f(x+0) = 2^x f(0) + 4^0 f(x) \]
This simplifies to:
\[ f(x) = 2^x f(0) + f(x) \]
From this, we can deduce:
\[ 0 = 2^x f(0) \]
Since \( 2^x \) is never zero for any real \( x \), we conclude that:
\[ f(0) = 0 \]

### Step 3: Substitute \( x = 1 \)
Now, substitute \( x = 1 \) into the original equation:
\[ f(1+y) = 2^1 f(y) + 4^y f(1) \]
This simplifies to:
\[ f(1+y) = 2 f(y) + 4^y \cdot 2 \]
Thus:
\[ f(1+y) = 2 f(y) + 2 \cdot 4^y \]

### Step 4: Substitute \( y = 1 \)
Next, substitute \( y = 1 \):
\[ f(x+1) = 2^x f(1) + 4^1 f(x) \]
This gives us:
\[ f(x+1) = 2^x \cdot 2 + 4 f(x) \]
So:
\[ f(x+1) = 2^{x+1} + 4 f(x) \]

### Step 5: Analyze the derived equations
Now we have two equations:
1. \( f(1+y) = 2 f(y) + 2 \cdot 4^y \)
2. \( f(x+1) = 2^{x+1} + 4 f(x) \)

### Step 6: Assume a form for \( f(x) \)
Assume \( f(x) = k \cdot 4^x + m \cdot 2^x \). We will find \( k \) and \( m \).

### Step 7: Substitute into the functional equation
Substituting \( f(x) = k \cdot 4^x + m \cdot 2^x \) into the original functional equation:
\[ f(x+y) = k \cdot 4^{x+y} + m \cdot 2^{x+y} \]
And the right side becomes:
\[ 2^x (k \cdot 4^y + m \cdot 2^y) + 4^y (k \cdot 4^x + m \cdot 2^x) \]
This leads to:
\[ k \cdot 4^{x+y} + m \cdot 2^{x+y} = k \cdot 2^x \cdot 4^y + m \cdot 2^x \cdot 2^y + k \cdot 4^y \cdot 4^x + m \cdot 4^y \cdot 2^x \]

### Step 8: Solve for coefficients
By equating coefficients, we can solve for \( k \) and \( m \). After some calculations, we find:
\[ f(x) = 4^x - 2^x \]

### Step 9: Find the minimum value of \( f(x) \)
Now, we need to find the minimum value of \( f(x) \):
\[ f(x) = 4^x - 2^x \]
To find the minimum, we can take the derivative and set it to zero:
\[ f'(x) = 4^x \ln(4) - 2^x \ln(2) = 0 \]
Solving this gives us the critical points.

### Step 10: Evaluate \( f(x) \) at critical points
After solving, we find that the minimum occurs at \( x = 0 \):
\[ f(0) = 4^0 - 2^0 = 1 - 1 = 0 \]

### Conclusion
Thus, the minimum value of \( f(x) \) is:
\[ \boxed{0} \]