If x and y are positive real numbers and xy = 8 , then the minimum value of 2 x + y is

Options

A. 9
B. 17
C. 10
D. 8

Answer

To find the minimum value of \(2x + y\) given that \(xy = 8\) and both \(x\) and \(y\) are positive real numbers, we can follow these steps:

### Step 1: Express \(y\) in terms of \(x\)
Since we know that \(xy = 8\), we can express \(y\) as:
\[ y = \frac{8}{x} \]

### Step 2: Substitute \(y\) into the expression \(2x + y\)
Now, we substitute \(y\) into the expression \(2x + y\):
\[ 2x + y = 2x + \frac{8}{x} \]

### Step 3: Find the derivative to locate critical points
To find the minimum value, we need to differentiate the expression \(2x + \frac{8}{x}\) with respect to \(x\) and set the derivative to zero:
\[ \frac{d}{dx}(2x + \frac{8}{x}) = 2 - \frac{8}{x^2} \]
Setting the derivative equal to zero gives:
\[ 2 - \frac{8}{x^2} = 0 \]
\[ \frac{8}{x^2} = 2 \]
\[ 8 = 2x^2 \]
\[ x^2 = 4 \]
\[ x = 2 \quad (\text{since } x \text{ is positive}) \]

### Step 4: Find the corresponding value of \(y\)
Now that we have \(x = 2\), we can find \(y\):
\[ y = \frac{8}{x} = \frac{8}{2} = 4 \]

### Step 5: Calculate the minimum value of \(2x + y\)
Now we can substitute \(x\) and \(y\) back into the expression \(2x + y\):
\[ 2x + y = 2(2) + 4 = 4 + 4 = 8 \]

### Conclusion
Thus, the minimum value of \(2x + y\) is:
\[ \boxed{8} \]

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