p and q are real numbers, and the equation x|x| = px + q has exactly k distinct real solutions for x.
Which one of the following is the complete list of possible values for k?
- A0, 1, 2
- B0, 1, 2, 3
- C0, 1, 2, 3, 4
- D0, 2, 4
- E1, 2, 3
- F1, 2, 3, 4
Show the answer and worked solution
answer · E
- A0, 1, 2
- B0, 1, 2, 3
- C0, 1, 2, 3, 4
- D0, 2, 4
- E1, 2, 3
- F1, 2, 3, 4
The curve y = x|x| is x2 for x ≥ 0 and −x2 for x < 0: continuous, odd, and strictly increasing from −∞ to +∞. So a line always meets it at least once, and k = 0 is impossible. On x ≥ 0 the equation is x2 − px − q = 0 and on x < 0 it is x2 + px + q = 0; two negative roots need q > 0, but then the first quadratic has product of roots −q < 0 and contributes only one, so k = 4 is impossible and k ≤ 3. All of 1, 2, 3 occur: p = q = 0 gives one; p = 2, q = −1 gives x = 1 and x = −1−√2; p = 4, q = 1 gives three.