If `f(x)` is a differentiable function satisfying `|f'(x)|le4AA 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 nore than 2 solutions in `x in [0, 4]`
C. `f(x)=14` has no solution in `x in [0, 4]`
D. `f(x)=20` has 2 solution in `xin[0, 4]`

Answer

To solve the problem, we need to analyze the differentiable function \( f(x) \) given the constraints on its derivative \( f'(x) \).

### Step 1: Understand the constraints on \( f'(x) \)
We know that:
\[ |f'(x)| \leq 4 \quad \text{for all } x \in [0, 4] \]
This means:
\[ -4 \leq f'(x) \leq 4 \]

### Step 2: Integrate the derivative to find \( f(x) \)
Since \( f'(x) \) is bounded, we can express \( f(x) \) in terms of its derivative:
\[ f(x) = f(0) + \int_0^x f'(t) \, dt \]
Given that \( f(0) = 0 \), we have:
\[ f(x) = \int_0^x f'(t) \, dt \]

### Step 3: Determine the bounds for \( f(x) \)
Using the bounds on \( f'(x) \):
- The maximum value of \( f'(t) \) is 4, thus:
\[ f(x) \leq \int_0^x 4 \, dt = 4x \]
- The minimum value of \( f'(t) \) is -4, thus:
\[ f(x) \geq \int_0^x (-4) \, dt = -4x \]

Combining these inequalities, we find:
\[ -4x \leq f(x) \leq 4x \]

### Step 4: Evaluate the bounds at the endpoints of the interval
Now we evaluate these bounds at the endpoints of the interval \( [0, 4] \):
- At \( x = 0 \):
\[ f(0) = 0 \]
- At \( x = 4 \):
\[ f(4) \leq 4 \cdot 4 = 16 \quad \text{and} \quad f(4) \geq -4 \cdot 4 = -16 \]

Thus, we conclude:
\[ f(x) \text{ ranges from } -16 \text{ to } 16 \text{ for } x \in [0, 4]. \]

### Step 5: Analyze the options
Now we check the provided options:
1. \( f(x) = 18 \): No solution, since \( 18 \) is outside the range \([-16, 16]\).
2. \( f(x) = 14 \): Has a solution, since \( 14 \) is within the range.
3. \( f(x) = 20 \): No solution, since \( 20 \) is outside the range.
4. \( f(x) = 14 \): Has a solution, since \( 14 \) is within the range.

### Conclusion
The only option that is correct is:
- **Option A**: \( f(x) = 18 \) has no solution in \( x \in [0, 4] \).