The range of the expression `2^(x) + 2^(-x) + 3^(x) + 3^(-x)` for `x in R`, is

Options

A. `(-oo, oo)`
B. `[4, oo)`
C. `(6, oo)`
D. `(8, oo)`

Answer

To find the range of the expression \(2^x + 2^{-x} + 3^x + 3^{-x}\) for \(x \in \mathbb{R}\), we can break down the expression into two parts and analyze them separately.

### Step 1: Rewrite the expression
We can rewrite the expression as:
\[ 2^x + 2^{-x} + 3^x + 3^{-x} \]
This can be grouped as:
\[ (2^x + 2^{-x}) + (3^x + 3^{-x}) \]

### Step 2: Analyze \(2^x + 2^{-x}\)
Let \(y = 2^x\). Then, we can express \(2^x + 2^{-x}\) as:
\[ y + \frac{1}{y} \]
We know from the AM-GM inequality that:
\[ y + \frac{1}{y} \geq 2 \]
This means that the minimum value of \(2^x + 2^{-x}\) is 2, which occurs when \(y = 1\) (i.e., when \(x = 0\)).

### Step 3: Analyze \(3^x + 3^{-x}\)
Similarly, let \(z = 3^x\). Then, we can express \(3^x + 3^{-x}\) as:
\[ z + \frac{1}{z} \]
Again, using the AM-GM inequality:
\[ z + \frac{1}{z} \geq 2 \]
This means that the minimum value of \(3^x + 3^{-x}\) is also 2, which occurs when \(z = 1\) (i.e., when \(x = 0\)).

### Step 4: Combine the results
Now, we combine the minimum values from both parts:
\[ (2^x + 2^{-x}) + (3^x + 3^{-x}) \geq 2 + 2 = 4 \]
Thus, the minimum value of the entire expression is 4.

### Step 5: Determine the maximum value
As \(x\) approaches positive or negative infinity, both \(2^x + 2^{-x}\) and \(3^x + 3^{-x}\) will approach infinity. Therefore, the maximum value of the expression can go to infinity.

### Conclusion
The range of the expression \(2^x + 2^{-x} + 3^x + 3^{-x}\) for \(x \in \mathbb{R}\) is:
\[ [4, \infty) \]