# f(x) = x^3 + cx Describe how the graph of fvaries as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how...

f(x) = x^3 + cx Describe how the graph of fvaries as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of c at which the basic shape of the curve changes.

lfryerda | Certified Educator

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.