The number of integers $k$ for which the equation $x^3-27x+k=0$ has at least two distict integer roots is ?  

a. 1 b. 2 c. 3 d. 4  

My attempt is to differentiate this cubic, I get a quadratic with real and distinct roots that is this cubic has a maximum and minimum, hence it cuts the $x$ axis always at three distinct points for any $k$.  

Now using Vieta's formulas I conclude that if two roots of the cubic have to be integers, k as well as the third root should be an integer ?  

Also there will be one root between $-3$ and $3$ i.e. points of maximum and minimum. Hence taking integer roots between them i.e. $1,2,-1,-2,0$, I get that for none of it I get integer roots for the remaining cubic.  

Hence no $k$ exists, i.e. $No.\; of\; k = 0$ which is incorrect.  

Where am I wrong or is there a simpler way to find ? Please use basic techniques of algebra to solve not with the help of very advanced topics.