Calculus: Early Transcendentals, Chapter 4, 4.6, Section 4.6, Problem 28

When c=0 , this is the basic cubic polynomial f(x)=x^3 which has no maximum, or minimum, and has horizontal tangent at x=0. Its graph is given by

There is also an inflection point as the function goes from concave down to concave up.
If c=-k<0 , then the function becomes f(x)=x^3-kx which has roots at x=0 and x=+-\sqrt k . The derivative is f'(x)=3x^2-k , and so we see there are local extrema at x=+-\sqrt{k/3}. The left extrema is a maximum and the right extrema is a minimum. An example of this graph is given by:

In addition to this, we see that the second derivative is f''(x)=6x and so there is an inflection point at x=0 .
Finally, we see that for c>0, there is a root only at x=0 . The derivative is f'(x)=3x^2+c , which never vanishes. Therefore, there is no maximum nor minimum for this function. There is an inflection point at x=0 since the second derivative is also f''(x)=6x . An example of a graph is given below.