`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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Chapter 4, 4.6 - Problem 28 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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lfryerda | High School Teacher | (Level 2) Educator

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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.

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