What is the complete range of values of k for which the curves with equations y = x3 − 12x and y = k − (x−2)2 intersect at three distinct points, of which exactly two have positive x-coordinates?
- A−4 < k < 0
- B−4 < k < 4
- C−4 < k < 16
- D−16 < k < 0
- E−16 < k < 4
- F−16 < k < 16
Show the answer and worked solution
answer · E
- A−4 < k < 0
- B−4 < k < 4
- C−4 < k < 16
- D−16 < k < 0
- E−16 < k < 4
- F−16 < k < 16
Set the two expressions equal: x3 − 12x = k − (x−2)2 rearranges to k = x3 + x2 − 16x + 4 = g(x), so intersections are where the horizontal line y = k meets the cubic g. Since g'(x) = 3x2 + 2x − 16 = (3x+8)(x−2), the local maximum is at x = −83 with g = 94027 and the local minimum is at x = 2 with g = −16; three distinct roots therefore need −16 < k < 94027. For the sign condition, the largest root always exceeds 2 and the smallest is always below −83, so everything hinges on the middle root, which lies on the decreasing branch through g(0) = 4: it is positive exactly when k < 4. Combining gives −16 < k < 4.