Multiple choice

The number of values of k for which the equation $\mathrm{x}^{3}-3\mathrm{x}+\mathrm{k}=0$ has two different roots lying in the interval (0,1) are

  1. 3

  2. 2

  3. infinitely many the requirement

  4. no value of k satisfies
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let f(x) = x^3 - 3x + k. For two roots in (0,1), we examine the derivative f'(x) = 3x^2 - 3. In (0,1), f'(x) < 0, so the function is strictly decreasing. A strictly decreasing function can have at most one root in an interval. Thus, it cannot have two different roots in (0,1).

AI explanation

Let f(x) = x^3 - 3x, making the derivative f'(x) = 3x^2 - 3. Setting the derivative to zero reveals local extrema at x = 1 and x = -1, meaning the function is strictly decreasing on the interval (0, 1). A strictly decreasing function on an interval is one-to-one, so it can intersect any horizontal line y = -k at exactly one point within that interval, making it impossible for the equation to have two different roots in (0, 1). Therefore, no value of k satisfies this requirement.

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