Question

The real number $\mathrm{K}$ for which the equation, $2{\mathrm{x}}^3+3\mathrm{x}+\mathrm{k}=0$ 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

Compute the value:

Let $\mathrm{f}\left(\mathrm{x}\right)=2{\mathrm{x}}^3+3\mathrm{x}+\mathrm{k}$

Differentiate with respect to $x$,

$\mathrm{f}\text{'}\left(\mathrm{x}\right)=6{\mathrm{x}}^2+3$

For all $x$ belongs to $\left[0,1\right]$ , $\mathrm{f}\text{'}\left(\mathrm{x}\right)>0$.

Since $\mathrm{f}\left(\mathrm{x}\right)$ is a strictly increasing function.

So, $\mathrm{f}\left(\mathrm{x}\right)=0$ has a single real root.

Therefore, two distinct real roots are impossible.

Hence option (D) is the correct answer.