Multiple choice

If 27a + 9b + 3c + d = 0, then the equation 4ax3 + 3bx2 + 2cx + d = 0 has at least one real root lying in the interval

  1. (–3, 0)

  2. (3, 4)

  3. (0, 3)

  4. (4, 5)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let f(x) = ax^4 + bx^3 + cx^2 + dx. Then f'(x) = 4ax^3 + 3bx^2 + 2cx + d. Given 27a + 9b + 3c + d = 0, this is f'(3) = 0. Also f(0) = 0 and f(3) = 81a + 27b + 9c + 3d = 3(27a + 9b + 3c + d) = 0. By Rolle's Theorem, there is a root of f'(x) in (0, 3).

AI explanation

Let f of x equal a times x to the fourth power plus b times x cubed plus c times x squared plus d times x. The derivative of this function is 4a times x cubed plus 3b times x squared plus 2c times x plus d. The given condition 27a plus 9b plus 3c plus d equals 0 implies f of 3 equals 0. Since f of 0 clearly equals 0, Rolle's Theorem guarantees that the derivative has at least one root in the interval from 0 to 3.

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