Answer: k = 9/5.
- A k = 1
- B k = 9/5
- C k = 5
- D k = 3
Correct answer: B. k = 9/5
Explanation: Continuity requires left limit = right limit = f(5). Left: lim(x→5⁻) = 5k+1. Right: lim(x→5⁺) = 3(5)−5 = 10. Setting 5k+1 = 10 gives 5k = 9, so k = 9/5.
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.