If `(d)/(dx) [f (x) ]=e^(-x) f(x) +e^x f(-x) `then `f (x) ` is (given f (0)=0)

Options

A. even
B. odd
C. neither even nor odd
D. can't say

Answer

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

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

We need to analyze this equation to find the function \( f(x) \) and determine whether it is even, odd, or neither. We also know that \( f(0) = 0 \).

### Step 1: Define a new function
Let's define a new function \( g(x) \) based on the right-hand side of the equation:

\[ g(x) = e^{-x} f(x) + e^{x} f(-x) \]

### Step 2: Find \( g(-x) \)
Next, we will find \( g(-x) \):

\[ g(-x) = e^{x} f(-x) + e^{-x} f(x) \]

### Step 3: Compare \( g(-x) \) and \( g(x) \)
Notice that:

\[ g(-x) = e^{x} f(-x) + e^{-x} f(x) = g(x) \]

This shows that \( g(x) \) is an even function because \( g(-x) = g(x) \).

### Step 4: Analyze the derivative
Since we have established that \( g(x) \) is even, we can use the property of derivatives. The derivative of an even function is odd. Therefore, we can conclude that:

\[ \frac{d}{dx}[f(x)] = g(x) \quad \text{(which is even)} \]

This means that \( f(x) \) must be an odd function because the derivative of an even function is odd.

### Step 5: Conclusion about \( f(x) \)
Since \( f(x) \) is odd, we can express this as:

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

Given that \( f(0) = 0 \), we can confirm that \( f(x) \) is indeed an odd function.

### Final Answer
Thus, \( f(x) \) is an odd function.

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