Answer: x = 0 and x = 2.
- A x = 0, treated as the single inflection point
- B x = 2, treated as the single inflection point
- C x = 0 and x = 2
- D x = 4, a value where the function is not actually inflecting
Correct answer: C. x = 0 and x = 2
Explanation: Compute f''(x): f'(x) = 4x³ − 12x², f''(x) = 12x² − 24x = 12x(x−2). f'' = 0 at x = 0 and x = 2. Sign of f'' changes at both points (−/+/−), confirming inflection points at x = 0 and x = 2.
As point Q slides along the curve toward P, the secant line PQ rotates into the tangent line at P, whose slope is the derivative.
Concept context
Rate of change, differentiation rules, and applications