Let f be an even function and f'(x) exists, then f'(0) is

Options

A. 1
B. 0
C. -1
D. -2

Answer

To solve the problem, we need to find the value of \( f'(0) \) given that \( f \) is an even function and \( f'(x) \) exists.

### Step-by-Step Solution:

1. **Understanding Even Functions**:
Since \( f \) is an even function, by definition, we have:
\[ f(x) = f(-x) \]
for all \( x \).

2. **Differentiating Both Sides**:
We differentiate both sides of the equation \( f(x) = f(-x) \) with respect to \( x \):
\[ \frac{d}{dx}[f(x)] = \frac{d}{dx}[f(-x)] \]

3. **Applying the Chain Rule**:
The left-hand side is simply \( f'(x) \). For the right-hand side, we apply the chain rule:
\[ f'(-x) \cdot (-1) = -f'(-x) \]
Therefore, we have:
\[ f'(x) = -f'(-x) \]

4. **Evaluating at \( x = 0 \)**:
Now, we substitute \( x = 0 \) into the equation we derived:
\[ f'(0) = -f'(0) \]

5. **Solving the Equation**:
Rearranging the equation gives:
\[ f'(0) + f'(0) = 0 \quad \Rightarrow \quad 2f'(0) = 0 \]
Dividing both sides by 2, we find:
\[ f'(0) = 0 \]

### Final Answer:
Thus, the value of \( f'(0) \) is \( 0 \).

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