Question

The number of values of $k$ for which the equation $x^3-3x+k=0$ has two distinct roots lying in the interval $(0,1)$ is

• A. three
• B. two
• C. infinitely many
• D. no value of $k$ satisfies the requirement.

Answer 1

Answer: (D)

no value of $k$ satisfies the requirement.

The given equation is:

$y=x^3-3x+k$

$\Rightarrow y^{\prime }=3x^2-3$

$y^{\prime }=0$

$\Rightarrow 3x^2-3=0$

$\Rightarrow x=\pm 1$. Thus the stationary points being $+1$ and $-1$ they lie outside the given region hence no value of k can make the equation roots lie in the given range of $(0,1)$. ....Answer