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

Let $f(x)=2x^3+3x+k$.$f^{\prime }(x)=6x^2+3$.Since $f^{\prime }(x)>0$ for all real $x$, the function $f(x)$ is strictly increasing.A strictly monotonic function can have at most one real root.Therefore, it is impossible for the equation to have two distinct real roots.