Question

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

• A. lies between 1 and 2
• B. lies between 2 and 3
• C. lies between -1 and 0
• D. does not exist.

Answer 1

The correct option is D does not exist.

$f\left(x\right)=2x^3+3x+k{f}^{\prime }\left(x\right)=6x^2+3>0\forall xϵR(\because x^2>0)$
$\Rightarrow f\left(x\right)$ is strictly increasing function
$\Rightarrow f\left(x\right)=0$ has only one real root, so two roots are not possible.