If a function `f: [-2a, 2a] -> R` is an odd function such that, `f(x) = f(2a - x)` for `x in [a, 2a]` and the left-hand derivative at `x = a` is `0`, then find the left-hand derivative at `x = -a`.

Answer

To solve the problem step-by-step, we need to analyze the given information about the function \( f \) and apply the definitions of derivatives and properties of odd functions.

### Step 1: Understand the given information
We are given that \( f: [-2a, 2a] \to \mathbb{R} \) is an odd function, which means:
\[ f(-x) = -f(x) \quad \text{for all } x. \]
We also know that \( f(x) = f(2a - x) \) for \( x \in [a, 2a] \) and that the left-hand derivative (LHD) at \( x = a \) is \( 0 \).

### Step 2: Write the expression for the left-hand derivative at \( x = a \)
The left-hand derivative at \( x = a \) is defined as:
\[ f'(a^-) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h}. \]
Given that \( f'(a^-) = 0 \), we can write:
\[ \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} = 0. \]

### Step 3: Write the expression for the left-hand derivative at \( x = -a \)
Now, we need to find the left-hand derivative at \( x = -a \):
\[ f'(-a^-) = \lim_{h \to 0^-} \frac{f(-a + h) - f(-a)}{h}. \]

### Step 4: Use the property of the odd function
Since \( f \) is an odd function, we have:
\[ f(-x) = -f(x). \]
Thus, we can express \( f(-a) \) as:
\[ f(-a) = -f(a). \]
Now, substituting \( -a + h \) into the function:
\[ f(-a + h) = f(-(a - h)) = -f(a - h). \]

### Step 5: Substitute into the derivative expression
Now we can substitute these into the left-hand derivative expression:
\[ f'(-a^-) = \lim_{h \to 0^-} \frac{-f(a - h) - (-f(a))}{h} = \lim_{h \to 0^-} \frac{-f(a - h) + f(a)}{h}. \]

### Step 6: Rearranging the expression
This can be rearranged as:
\[ f'(-a^-) = -\lim_{h \to 0^-} \frac{f(a - h) - f(a)}{h}. \]

### Step 7: Recognize the limit
Notice that the limit we have now is the definition of the left-hand derivative at \( x = a \):
\[ f'(-a^-) = -f'(a^-). \]
Since we know \( f'(a^-) = 0 \):
\[ f'(-a^-) = -0 = 0. \]

### Conclusion
Thus, the left-hand derivative at \( x = -a \) is:
\[ \boxed{0}. \]