Question

Hi, this question was in a year 11 extension maths textbook in the enrichment section. I have the answer as k>17 and k<-11 because I graphed it on GeoGebra. The Graph can be found here: https://ggbm.at/xpegwwtq. While I know the answers I would like to know how to work it out using algebra.

Here is the Question:
Consider the cubic equation x^3 - kx + (k + 11) = 0, find all the integer values of k for which the equation has at least one positive integer solution for x

Thanks, Taran