Show that, for any cubic function of the form y= ax^3+bx^2+cx+d there is a single point of inflection, and the slope of the curve at that point c-(b^2/3a)

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rcmath eNotes educator| Certified Educator

To find the point of inflection, we need to find the second derivative.

`y=ax^3+bx^2+cx+d =>`




Since we are discussing cubic function, the assumption is that a is not zero. Thus for any cubic function of the given form, we have the inflection point at `x=-b/(3a)`

To find the slope at this point, we need to plug in the value of x in the first derivative function.




Thus the slope is `-(b^2)/(3a)+c`