Question

If \[ f(x) = |x - 2|, \quad x \in [0, 4], \quad \text{then the Rolle's theorem cannot be applied to the function because} \]

• A. The function is not differentiable at every point in the \( (0, 4) \)
• B. \( f(4) \neq f(0) \)
• C. Function is not well-defined in the domain
• D. The function is not continuous at every point in the \( [0, 4] \)

Hint:

Rolle's theorem requires the function to be differentiable on the open interval and continuous on the closed interval. If there is a sharp corner or cusp, the function is not differentiable at that point.

Answer 1

The Correct Option is A

Step 1: Apply Rolle's Theorem.
Rolle’s theorem can be applied if the function is continuous on the closed interval \( [a, b] \), differentiable on the open interval \( (a, b) \), and if \( f(a) = f(b) \). The function \( f(x) = |x - 2| \) is continuous on \( [0, 4] \), but it is not differentiable at \( x = 2 \), because at this point, the function has a sharp corner.
Step 2: Conclusion.
Since the function is not differentiable at \( x = 2 \), Rolle’s theorem cannot be applied. Therefore, the correct answer is option (A).