# `f(x) = x^3 + cx` Describe how the graph of `f`varies 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 `f`varies 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.

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### 1 Answer

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.