If `f(x) + f(Y) = f(x+y) and int_(0)^(3) f(x) dx= lamda`, then `int_(-3)^(3) f(x) dx`=

Options

A. `-2k`
B. 2k
C. 0
D. k/2

Answer

To solve the problem step by step, we will analyze the given functional equation and then evaluate the definite integral.

### Step 1: Analyze the functional equation
We are given the equation:
\[ f(x) + f(y) = f(x+y) \]
This is a functional equation that resembles the Cauchy functional equation.

### Step 2: Substitute specific values
1. Let \( y = 0 \):
\[ f(x) + f(0) = f(x + 0) \implies f(x) + f(0) = f(x) \]
This implies:
\[ f(0) = 0 \]

2. Now, let \( y = -x \):
\[ f(x) + f(-x) = f(x + (-x)) \implies f(x) + f(-x) = f(0) \implies f(x) + f(-x) = 0 \]
This implies:
\[ f(-x) = -f(x) \]
Hence, \( f(x) \) is an odd function.

### Step 3: Use the property of definite integrals
For any odd function \( f(x) \):
\[ \int_{-a}^{a} f(x) \, dx = 0 \]
In our case, we need to evaluate:
\[ \int_{-3}^{3} f(x) \, dx \]

### Step 4: Conclusion
Since \( f(x) \) is an odd function, we can conclude:
\[ \int_{-3}^{3} f(x) \, dx = 0 \]

### Final Answer
\[ \int_{-3}^{3} f(x) \, dx = 0 \]

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