If `f(2+x)=f(-x)` for all `x in R` then differentiability at x=4 implies differentiability at (a) x=1 (b) x=-1 (c) x=-2 (d) cannot say anything

Options

A. x=1
B. x=-1
C. x=-2
D. cannot say anything

Answer

To solve the problem, we start with the given functional equation:

\[ f(2 + x) = f(-x) \]

for all \( x \in \mathbb{R} \).

### Step 1: Rewrite the functional equation
We can rewrite the equation by substituting \( x \) with \( -x \):

\[ f(2 - x) = f(x) \]

This shows that \( f \) is symmetric about the line \( x = 1 \) (since \( 2 - x \) reflects \( x \) around \( 1 \)).

### Step 2: Analyze differentiability at \( x = 4 \)
We need to find out what differentiability at \( x = 4 \) implies. We know that if \( f \) is differentiable at \( x = 4 \), then the derivative \( f'(4) \) exists.

### Step 3: Use the functional equation to find relationships
From the original equation, we can express the derivative at \( x = 4 \):

Using the chain rule, we differentiate \( f(2 + x) = f(-x) \):

\[ \frac{d}{dx} f(2 + x) = \frac{d}{dx} f(-x) \]

This gives us:

\[ f'(2 + x) = -f'(-x) \]

### Step 4: Substitute \( x = 2 \)
Now, substituting \( x = 2 \):

\[ f'(2 + 2) = -f'(-2) \]

This simplifies to:

\[ f'(4) = -f'(-2) \]

### Step 5: Analyze differentiability at \( x = -2 \)
From the relationship we found, if \( f'(4) \) exists, then \( -f'(-2) \) must also exist, which implies that \( f'(-2) \) exists. Thus, differentiability at \( x = 4 \) implies differentiability at \( x = -2 \).

### Conclusion
Therefore, the answer is:

(c) \( x = -2 \)