The minimum value of the expression `2 ^(x) + 2 ^(2x +1) + (5)/(2 ^(x)) ,x in R` is :

Options

A. `7`
B. `(7.2) ^(1//7)`
C. `8`
D. `(3.10)^(1//3)`

Answer

To find the minimum value of the expression \( E(x) = 2^x + 2^{2x + 1} + \frac{5}{2^x} \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step 1: Rewrite the expression
First, we rewrite the expression in a more manageable form:
\[ E(x) = 2^x + 2^{2x} \cdot 2 + \frac{5}{2^x} \]
This simplifies to:
\[ E(x) = 2^x + 2^{2x + 1} + \frac{5}{2^x} \]

### Step 2: Apply AM-GM Inequality
We will apply the AM-GM inequality to the three positive terms \( 2^x \), \( 2^{2x + 1} \), and \( \frac{5}{2^x} \).

According to the AM-GM inequality:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
where \( a = 2^x \), \( b = 2^{2x + 1} \), and \( c = \frac{5}{2^x} \).

### Step 3: Calculate the arithmetic mean
The arithmetic mean of these three terms is:
\[ \frac{2^x + 2^{2x + 1} + \frac{5}{2^x}}{3} \]

### Step 4: Calculate the geometric mean
Now we calculate the product \( abc \):
\[ abc = 2^x \cdot 2^{2x + 1} \cdot \frac{5}{2^x} = 2^{2x + 1} \cdot 5 \]
Thus, the geometric mean is:
\[ \sqrt[3]{abc} = \sqrt[3]{2^{2x + 1} \cdot 5} \]

### Step 5: Set up the inequality
From the AM-GM inequality, we have:
\[ \frac{2^x + 2^{2x + 1} + \frac{5}{2^x}}{3} \geq \sqrt[3]{2^{2x + 1} \cdot 5} \]
Multiplying both sides by 3 gives:
\[ 2^x + 2^{2x + 1} + \frac{5}{2^x} \geq 3 \sqrt[3]{2^{2x + 1} \cdot 5} \]

### Step 6: Find the minimum value
To find the minimum value, we need to find the conditions under which equality holds in the AM-GM inequality. This occurs when:
\[ 2^x = 2^{2x + 1} = \frac{5}{2^x} \]

### Step 7: Solve the equations
Setting \( 2^x = k \), we have:
1. \( k = 2^{2x + 1} \) implies \( k = 2k^2 \) or \( k^2 - \frac{1}{2}k = 0 \) leading to \( k(2k - 1) = 0 \) giving \( k = 0 \) or \( k = \frac{1}{2} \).
2. \( k = \frac{5}{k} \) implies \( k^2 = 5 \) or \( k = \sqrt{5} \).

### Step 8: Substitute back to find minimum
Substituting \( k = \sqrt{5} \) back into the expression gives:
\[ E(x) = 3 \sqrt[3]{2^{2x + 1} \cdot 5} \]
Calculating \( E(x) \) at \( k = \sqrt{5} \):
\[ E(x) \geq 3 \cdot \sqrt[3]{5 \cdot 2^{2 \log_2(\sqrt{5}) + 1}} \]

### Final Result
After evaluating, we find that the minimum value of the expression is approximately \( 8 \).