The smallest natural value of a for which the function `f(x)=2(x+1)-a(2^(-x))+(2a+1)(ln2)x-6` is increasing `AA x in R`, is

Answer

To find the smallest natural value of \( a \) for which the function

\[ f(x) = 2(x + 1) - a(2^{-x}) + (2a + 1)(\ln 2)x - 6 \]

is increasing for all \( x \in \mathbb{R} \), we need to analyze the derivative of the function. A function is increasing if its derivative is non-negative for all \( x \).

### Step 1: Find the derivative of \( f(x) \)

We start by differentiating \( f(x) \):

\[ f'(x) = \frac{d}{dx}\left[2(x + 1)\right] - \frac{d}{dx}\left[a(2^{-x})\right] + \frac{d}{dx}\left[(2a + 1)(\ln 2)x\right] - \frac{d}{dx}[6] \]

Calculating each term:

1. The derivative of \( 2(x + 1) \) is \( 2 \).
2. The derivative of \( -a(2^{-x}) \) is \( a(2^{-x} \ln(2)) \) (using the chain rule).
3. The derivative of \( (2a + 1)(\ln 2)x \) is \( (2a + 1)(\ln 2) \).
4. The derivative of \( -6 \) is \( 0 \).

Thus, we have:

\[ f'(x) = 2 + a(2^{-x} \ln 2) + (2a + 1)(\ln 2) \]

### Step 2: Set the derivative greater than or equal to zero

For \( f(x) \) to be increasing, we need:

\[ f'(x) \geq 0 \]

This leads to:

\[ 2 + a(2^{-x} \ln 2) + (2a + 1)(\ln 2) \geq 0 \]

### Step 3: Rearranging the inequality

Rearranging gives us:

\[ a(2^{-x} \ln 2) + (2a + 1)(\ln 2) \geq -2 \]

### Step 4: Analyze the terms

Notice that \( 2^{-x} \) decreases as \( x \) increases. Thus, the term \( a(2^{-x} \ln 2) \) can become very small for large \( x \). Therefore, we need to ensure that the inequality holds for all \( x \).

### Step 5: Find the critical point

To find the minimum value of \( a \), we can analyze the situation when \( x \) approaches infinity, where \( 2^{-x} \) approaches \( 0 \):

\[ (2a + 1)(\ln 2) \geq -2 \]

This simplifies to:

\[ 2a + 1 \geq -\frac{2}{\ln 2} \]

### Step 6: Solve for \( a \)

Calculating \( -\frac{2}{\ln 2} \):

Using \( \ln 2 \approx 0.693 \):

\[ -\frac{2}{\ln 2} \approx -\frac{2}{0.693} \approx -2.886 \]

Thus, we have:

\[ 2a + 1 \geq -2.886 \]

Subtracting \( 1 \):

\[ 2a \geq -3.886 \]

Dividing by \( 2 \):

\[ a \geq -1.943 \]

### Step 7: Finding the smallest natural number

Since \( a \) must be a natural number, the smallest natural number satisfying this condition is \( 1 \).

### Conclusion

The smallest natural value of \( a \) for which the function \( f(x) \) is increasing for all \( x \in \mathbb{R} \) is:

\[ \boxed{1} \]