If `f(x)=mx+c`,`f(0)=f^`(0)=1` then `f(2)=`

Options

A. 1
B. 2
C. 3
D. -3

Explanation

To solve the problem, we need to find the value of \( f(2) \) given the function \( f(x) = mx + c \) and the conditions \( f(0) = 1 \) and \( f'(0) = 1 \).

### Step-by-Step Solution:

1. **Identify the function**:
The function is given as \( f(x) = mx + c \).

2. **Use the first condition**:
We know that \( f(0) = 1 \).
Substituting \( x = 0 \) into the function:
\[ f(0) = m(0) + c = c \]
Therefore, we have:
\[ c = 1 \]

3. **Use the second condition**:
The second condition states that \( f'(0) = 1 \).
First, we need to find the derivative \( f'(x) \):
\[ f'(x) = m \]
Since \( f'(0) = m \), we have:
\[ m = 1 \]

4. **Write the complete function**:
Now that we have both \( m \) and \( c \), we can write the complete function:
\[ f(x) = 1x + 1 = x + 1 \]

5. **Calculate \( f(2) \)**:
Now we need to find \( f(2) \):
\[ f(2) = 2 + 1 = 3 \]

### Final Answer:
Thus, the value of \( f(2) \) is \( 3 \).

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