If `f(x)=sqrt(x+2sqrt(2x-4))+sqrt(x-2sqrt(2x-4))` then the value of 10 `f'(102^(+))`, is

Options

A. -1
B. 0
C. 1
D. does not exist

Answer

To solve the problem step by step, we start with the function given:

**Step 1: Define the function**
Given:
\[ f(x) = \sqrt{x + 2\sqrt{2x - 4}} + \sqrt{x - 2\sqrt{2x - 4}} \]

**Step 2: Simplify the function**
We can rewrite the terms inside the square roots. Notice that:
\[ 2\sqrt{2x - 4} = 2\sqrt{2(x - 2)} \]
Thus, we can express the function as:
\[ f(x) = \sqrt{x + 2\sqrt{2}\sqrt{x - 2}} + \sqrt{x - 2\sqrt{2}\sqrt{x - 2}} \]

Let \( y = \sqrt{x - 2} \), so \( x = y^2 + 2 \). Then, we can rewrite \( f(x) \) in terms of \( y \):
\[ f(y) = \sqrt{(y^2 + 2) + 2\sqrt{2}y} + \sqrt{(y^2 + 2) - 2\sqrt{2}y} \]

**Step 3: Further simplify the expression**
The first term simplifies to:
\[ \sqrt{(y + \sqrt{2})^2} = y + \sqrt{2} \]
The second term simplifies to:
\[ \sqrt{(y - \sqrt{2})^2} = y - \sqrt{2} \]

Thus, we have:
\[ f(y) = (y + \sqrt{2}) + (y - \sqrt{2}) = 2y \]

Since \( y = \sqrt{x - 2} \), we can write:
\[ f(x) = 2\sqrt{x - 2} \]

**Step 4: Differentiate the function**
Now, we differentiate \( f(x) \):
\[ f'(x) = 2 \cdot \frac{1}{2\sqrt{x - 2}} = \frac{1}{\sqrt{x - 2}} \]

**Step 5: Evaluate the derivative at \( x = 102 \)**
Now, we need to find \( f'(102) \):
\[ f'(102) = \frac{1}{\sqrt{102 - 2}} = \frac{1}{\sqrt{100}} = \frac{1}{10} \]

**Step 6: Calculate \( 10f'(102) \)**
Finally, we calculate:
\[ 10f'(102) = 10 \cdot \frac{1}{10} = 1 \]

Thus, the final answer is:
\[ \boxed{1} \]