Question

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

• A. lies between $1and2$
• B. lies between $2and3$
• C. lies between $-1and0$
• D. $doesnotexist$

Answer 1

The correct option is D $doesnotexist$

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

Differentiating above equation w.r.t. $x$, we get

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

$\Rightarrow f^{\prime }\left(x\right)>0$ for all $x\in R$

$\Rightarrow f^{\prime }\left(x\right)=3(2x^2+1)>0$

$\Rightarrow x>\frac{-1}{2}$ which is not possible

As condition for two distinct real roots is $f\left(\alpha \right)f\left(\beta \right)=0$

$f\left(x\right)$ is strictly increasing.

$\therefore$ No value of $k$ exist.