JEE Main · 2013 · 07 Apr 2013 · Offline · Application of Derivatives

For a cubic to carry two distinct roots in an interval, its derivative must vanish somewhere inside, by Rolle's theorem. Here $f(x)=2x^3+3x+k$ has $f^{\prime }(x)=6x^2+3$, which is strictly positive everywhere, so $f$ is strictly increasing on $[0,1]$ and can cross zero at most once. No value of $k$ can give two distinct real roots in $[0,1]$.