Answer: c = 2.
- A c = 1
- B c = 2
- C c = 3
- D c = 1.5
Correct answer: B. c = 2
Explanation: f(1) = 0 = f(3), satisfying the hypothesis. f'(x) = 2x - 4 = 0 gives x = 2, which lies in (1,3).
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.