$\text{ If the graph of }f\left(x\right)=2x^3+a{x}^2+bx(a,b\in N)\text{ cuts the }x\text{-axis at three distinct points, then minimum value of }(a+b)\text{ is }$

$\begin{array}{r}f\left(x\right)=2x^3+a{x}^2+bx;a,b\in N\\ =x\left(2x^2+ax+b\right)\end{array}$

$\text{ Given }f\left(x\right)\text{ cuts }x\text{-axis at }3\text{ distinct points then }$

$2x^2+ax+b\text{ must have }2\text{ distinct real roots }$

$\begin{array}{r}\therefore a^2-8b>0\\ \Rightarrow a^2>8b\end{array}$

$\text{ Since }a,b\in N\text{, minimum value of }a+b\text{ is }4\text{ when }a=3\text{ and b=1}$