Editing Cubic function (section)

==Critical and inflection points==
{{Cubic_graph_special_points.svg}}
The [[critical point (mathematics)|critical points]] of a cubic function are its [[stationary point]]s, that is the points where the slope of the function is zero.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Stationary Point|url=https://mathworld.wolfram.com/StationaryPoint.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref> Thus the critical points of a cubic function {{math|''f''}} defined by 
:{{math|''f''(''x'') {{=}} ''ax''<sup>3</sup> + ''bx''<sup>2</sup> + ''cx'' + ''d''}}, 
occur at values of {{math|''x''}} such that the [[derivative]]
:<math> 3ax^2 + 2bx + c = 0</math>
of the cubic function is zero.

The solutions of this equation are the {{mvar|x}}-values of the critical points and are given, using the [[quadratic formula]], by <!-- Do not change 3ac into 4ac: here the of the cubic equation coefficients of the quadratic polynomial are not the same as the coefficients generally used for expressing the quadratic formula -->
:<math>x_\text{critical}=\frac{-b \pm \sqrt {b^2-3ac}}{3a}.</math>

The sign of the expression {{math|Δ<sub>0</sub> {{=}} }}{{math|''b''{{sup|2}} − 3''ac''}} inside the square root determines the number of critical points. If it is positive, then there are two critical points, one is a local maximum, and the other is a local minimum. If {{math|''b''{{sup|2}} − 3''ac'' {{=}} 0}}, then there is only one critical point, which is an [[inflection point]]. If {{math|''b''{{sup|2}} − 3''ac'' < 0}}, then there are no (real) critical points. In the two latter cases, that is, if {{math|''b''{{sup|2}} − 3''ac''}} is nonpositive, the cubic function is strictly [[monotonic]]. See the figure for an example of the case {{math|Δ<sub>0</sub> > 0}}.

The inflection point of a function is where that function changes [[Second derivative#Concavity|concavity]].<ref>{{Cite book|last1=Hughes-Hallett|first1=Deborah|url=https://books.google.com/books?id=8CeVDwAAQBAJ&q=inflection+point+of+a+function+is+where+that+function+changes+concavity&pg=PA181|title=Applied Calculus|last2=Lock|first2=Patti Frazer|last3=Gleason|first3=Andrew M.|last4=Flath|first4=Daniel E.|last5=Gordon|first5=Sheldon P.|last6=Lomen|first6=David O.|last7=Lovelock|first7=David|last8=McCallum|first8=William G.|last9=Osgood|first9=Brad G.|date=2017-12-11|publisher=John Wiley & Sons|isbn=978-1-119-27556-5|pages=181|language=en|quote=A point at which the graph of the function f changes concavity is called an inflection point of f}}</ref> An inflection point occurs when the [[second derivative]] <math>f''(x) = 6ax + 2b, </math> is zero, and the third derivative is nonzero. Thus a cubic function has always a single inflection point, which occurs at
:<math>x_\text{inflection} = -\frac{b}{3a}.</math>