Multiple choice

If the equation $2t^{3}-9t^{2}+30-a=0$ has three real and distinct roots, then

  1. $a > 30$
  2. $a < 3$
  3. $3 < a < 30$
  4. $a < 3 \ or \ a > 30$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let f(t) = 2t^3 - 9t^2 + 30. For three real roots, the local maximum and minimum must straddle the value 'a'. f'(t) = 6t^2 - 18t = 6t(t-3). Critical points at t=0 and t=3. f(0) = 30 (local max), f(3) = 2(27) - 9(9) + 30 = 54 - 81 + 30 = 3 (local min). For three roots, 3 < a < 30.

AI explanation

Let f(t) = 2t^3 - 9t^2 + 30. The derivative f'(t) = 6t^2 - 18t = 0 gives the local extrema at t = 0 and t = 3. To have three distinct real roots for f(t) - a = 0, the line y = a must lie strictly between the local maximum f(0) = 30 and the local minimum f(3) = 3. Therefore, 3 < a < 30.

Keep practicing — related questions