If it is know that at the point x = 1 two anti- derivatives of `f(x) = e^(x)` differ by 2, the difference of the anti - derivatives at x = 100 is :

Options

A. `e^(100)`
B. 100
C. 2
D. None of these

Answer

To solve the problem, we need to find the difference between two antiderivatives of the function \( f(x) = e^x \) at \( x = 100 \), given that they differ by 2 at \( x = 1 \).

### Step-by-Step Solution:

1. **Identify the Antiderivatives**:
The antiderivative of \( f(x) = e^x \) is given by:
\[ F(x) = e^x + C \]
where \( C \) is an arbitrary constant.

2. **Define Two Antiderivatives**:
Let the two antiderivatives be:
\[ F_1(x) = e^x + c_1 \]
\[ F_2(x) = e^x + c_2 \]
where \( c_1 \) and \( c_2 \) are constants.

3. **Use the Given Condition**:
We know that at \( x = 1 \), the difference between the two antiderivatives is 2:
\[ F_1(1) - F_2(1) = 2 \]
Substituting the expressions for \( F_1 \) and \( F_2 \):
\[ (e^1 + c_1) - (e^1 + c_2) = 2 \]
Simplifying this gives:
\[ c_1 - c_2 = 2 \quad \text{(Equation 1)} \]

4. **Find the Difference at \( x = 100 \)**:
Now we need to find the difference between the two antiderivatives at \( x = 100 \):
\[ F_1(100) - F_2(100) = (e^{100} + c_1) - (e^{100} + c_2) \]
Simplifying this gives:
\[ F_1(100) - F_2(100) = c_1 - c_2 \]

5. **Substitute from Equation 1**:
From Equation 1, we know that:
\[ c_1 - c_2 = 2 \]
Therefore:
\[ F_1(100) - F_2(100) = 2 \]

### Final Answer:
The difference of the antiderivatives at \( x = 100 \) is:
\[ \boxed{2} \]