Multiple choice

The real number k for which the equation, 2x3 + 3x + k = 0 has two distinct real roots in [0, 1]

  1. lies between 2 and 3

  2. lies between – 1 and 0

  3. does not exist

  4. lies between 1 and 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let f(x) = 2x^3 + 3x + k. The derivative f'(x) = 6x^2 + 3 is always positive, meaning the function is strictly increasing. A strictly increasing function can have at most one real root, so it cannot have two distinct real roots in [0, 1].

AI explanation

The derivative of f(x) = 2x^3 + 3x + k is f'(x) = 6x^2 + 3. Since f'(x) is strictly positive for all real x, the function is strictly monotonic, meaning it can cross the x-axis at most once. Therefore, a cubic equation of this form can never have two distinct real roots in any interval.

Keep practicing — related questions