Let f be a differentiable function defined on R such that f(0) =`-3`. If f'(x) `le 5` for all x then

Options

A. `f(2) gt 7`
B. `f(2) le 7`
C. `f(2) gt 8`
D. `f(2) = 8`

Answer

To solve the problem, we will use the information given about the differentiable function \( f \) and its derivative \( f' \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We know that \( f(0) = -3 \).
- We also know that \( f'(x) \leq 5 \) for all \( x \).

2. **Use the Mean Value Theorem**:
- According to the Mean Value Theorem, if \( f \) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one \( c \in (a, b) \) such that:
\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]
- We can apply this theorem to the interval \([0, x]\) for any \( x > 0 \).

3. **Apply the Mean Value Theorem**:
- Let \( a = 0 \) and \( b = x \). Then we have:
\[ f'(c) = \frac{f(x) - f(0)}{x - 0} = \frac{f(x) + 3}{x} \]
- Since \( f'(x) \leq 5 \), we can say:
\[ \frac{f(x) + 3}{x} \leq 5 \]

4. **Rearranging the Inequality**:
- Multiply both sides by \( x \) (assuming \( x > 0 \)):
\[ f(x) + 3 \leq 5x \]
- Subtract 3 from both sides:
\[ f(x) \leq 5x - 3 \]

5. **Evaluate at \( x = 2 \)**:
- Now, we want to find \( f(2) \):
\[ f(2) \leq 5(2) - 3 = 10 - 3 = 7 \]
- Therefore, we conclude:
\[ f(2) \leq 7 \]

### Final Result:
Thus, the value of \( f(2) \) is less than or equal to 7.