If f : R - R is an even function which is twice differentiable on R and `f''(pi)=1`, then `f''(-pi)`

Options

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

Answer

To solve the problem, we need to utilize the properties of even functions and the information given about the second derivative.

### Step-by-step Solution:

1. **Understanding Even Functions**:
An even function \( f(x) \) satisfies the property:
\[ f(x) = f(-x) \quad \text{for all } x \in \mathbb{R} \]

2. **Differentiating the Even Function**:
If \( f(x) \) is an even function, then its first derivative \( f'(x) \) is an odd function. This means:
\[ f'(-x) = -f'(x) \]

3. **Differentiating Again**:
The second derivative \( f''(x) \) of an even function is also an even function. Thus, we have:
\[ f''(-x) = f''(x) \]

4. **Applying the Given Information**:
We are given that \( f''(\pi) = 1 \). Since \( f''(x) \) is even, we can write:
\[ f''(-\pi) = f''(\pi) \]

5. **Substituting the Known Value**:
From the previous step, substituting the known value:
\[ f''(-\pi) = f''(\pi) = 1 \]

### Conclusion:
Thus, we conclude that:
\[ f''(-\pi) = 1 \]