Answer: c = pi/2.
- A c = 0
- B c = pi/2
- C c = pi/4
- D c = pi
Correct answer: B. c = pi/2
Explanation: Rolle's theorem conditions: sin x is continuous on [0,π], differentiable on (0,π), and f(0) = sin 0 = 0 = sin π = f(π). So theorem applies. f'(x) = cos x = 0 at c = π/2 ∈ (0,π).
f(x)=|x| is perfectly continuous at x=0 (no break, no jump), but the curve has a sharp corner there: approaching from the left gives slope -1 and from the right gives slope +1 - since these one-sided derivatives disagree, the function is not differentiable at that point.
Concept context
When a function has no breaks (continuity) and when it has a well-defined slope (differentiability), plus rules for differentiating implicit, inverse trig, exponential, log, and parametric functions.