If `f(x)` is a differentiable function satisfying `|f'(x)| le 2 AA x in [0, 4]` and `f(0)=0`, then

Options

A. `f(x)=18` has no solution in `x in [0, 4]`
B. `f(x)=18` has more than 2 solutions in `x in [0,4]`
C. `f(x)=14` has 3 solutions in `x in [0, 4]`
D. `f(x)=20` has 2 solutions in `x in [0, 4]`

Answer

To solve the problem, we start with the information given about the differentiable function \( f(x) \):

1. **Given Conditions**:
- \( |f'(x)| \leq 2 \) for \( x \in [0, 4] \)
- \( f(0) = 0 \)

2. **Understanding the Derivative Condition**:
The condition \( |f'(x)| \leq 2 \) implies that the rate of change of the function \( f(x) \) is bounded. Specifically, this means that the function can increase at most at a rate of 2 units per unit of \( x \) and decrease at most at a rate of -2 units per unit of \( x \).

3. **Finding the Maximum and Minimum Values**:
- Since \( f'(x) \leq 2 \), the maximum increase in \( f(x) \) from \( x = 0 \) to \( x = 4 \) can be calculated as:
\[ f(4) - f(0) \leq 2 \cdot (4 - 0) = 8 \]
Thus, we have:
\[ f(4) \leq 8 \]
- Similarly, since \( f'(x) \geq -2 \), the minimum decrease in \( f(x) \) can be calculated as:
\[ f(4) - f(0) \geq -2 \cdot (4 - 0) = -8 \]
Thus, we have:
\[ f(4) \geq -8 \]

4. **Combining the Results**:
From the above calculations, we conclude that:
\[ -8 \leq f(4) \leq 8 \]

5. **Conclusion**:
The possible values of \( f(4) \) lie within the range of \([-8, 8]\). Therefore, we check the options given in the question:
- \( f(x) = 18 \): Not possible since \( 18 > 8 \)
- \( f(x) = 14 \): Not possible since \( 14 > 8 \)
- \( f(x) = 20 \): Not possible since \( 20 > 8 \)

Thus, none of the values 18, 14, or 20 can be achieved by \( f(x) \) within the given constraints.

### Final Answer:
None of the options (18, 14, 20) are valid for \( f(x) \).