Answer: f(a) = f(b).
- A f(a) > f(b)
- B f(a) = f(b)
- C f(a) < f(b)
- D f(a) times f(b) = 0
Correct answer: B. f(a) = f(b)
Explanation: Rolle's Theorem requires f continuous on [a,b], differentiable on (a,b), and f(a) = f(b); then some c in (a,b) has f'(c) = 0.
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.