The value of$k$ so that the equation  $x^3-12x+k=0$ has distinct roots in [0, 2] is

If $\alpha$ and $\beta$ are real and distinct roots of $x^3-$$12x+k=0.$ Then $f\left(x\right)=x^3-12x+k$ satisfies 

$f\left(\alpha \right)=f\left(\beta \right)=0$then there $\gamma \in [\alpha ,\beta ]\subseteq [0,2]$ such that

$f^{\mathrm{\prime }}\left(\gamma \right)=0$i.e., $3v^2-12=0\Rightarrow \gamma =\pm 2$. Not possible.