Let `f:R to R ` be a function satisfying `f(2-x)=f(2+x) and f(20-x)=f(x) AA x in R.` For this function `f`, answer the following.
If `f(2) ne f(6),` then the

Options

A. (a) fundamental period of `f(x) " is " 1`
B. (b) fundamental period of `f(x) " may be " 1`
C. (c) period of `f(x)` cannot be 1
D. (d) fundamental period of `f(x) " is " 8`

Explanation

To solve the problem, we need to analyze the given functional equations step by step.

### Step 1: Analyze the first functional equation
We start with the equation:
\[ f(2 - x) = f(2 + x) \]
This equation implies that the function \( f \) is symmetric about \( x = 2 \).

### Step 2: Substitute \( x \) with \( 2 - x \)
Let's replace \( x \) in the first equation with \( 2 - x \):
\[ f(2 - (2 - x)) = f(2 + (2 - x)) \]
This simplifies to:
\[ f(x) = f(4 - x) \]
This means that the function is also symmetric about \( x = 4 \).

### Step 3: Analyze the second functional equation
Now, consider the second functional equation:
\[ f(20 - x) = f(x) \]
This indicates that the function \( f \) is symmetric about \( x = 10 \).

### Step 4: Combine the results
From the first equation, we have:
1. \( f(x) = f(4 - x) \)
2. \( f(x) = f(20 - x) \)

### Step 5: Find the period of the function
We can use the symmetry properties to find the period of the function. From \( f(x) = f(20 - x) \), if we replace \( x \) with \( x + 20 \):
\[ f(x + 20) = f(20 - (x + 20)) = f(-x) \]
But since \( f(x) = f(20 - x) \), this means:
\[ f(x + 20) = f(x) \]
Thus, the function has a period of 20.

### Step 6: Check for smaller periods
Next, we need to check if there is a smaller period. We know:
- From \( f(x) = f(4 - x) \), we can check if there is a period of 8:
\[ f(x + 8) = f(4 - (x + 8)) = f(-4 - x) \]
This does not simplify back to \( f(x) \).

### Conclusion
Since we have established that the function is symmetric about \( x = 10 \) and \( x = 4 \), and we found that the period is 20, we conclude that the period of the function \( f \) is 20.

### Final Answer
Thus, if \( f(2) \neq f(6) \), then the period of the function \( f \) is:
\[ \text{Period} = 20 \]

---