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Question Number 91460 by  M±th+et+s last updated on 30/Apr/20
one of the conditions of the inflection point is inflection tangent. what is inflection tangent?
$${one}\:{of}\:{the}\:{conditions}\:{of}\:{the}\:{inflection} \\ $$$${point}\:{is}\:{inflection}\:{tangent}. \\ $$$${what}\:{is}\:{inflection}\:{tangent}? \\ $$
Answered by MJS last updated on 01/May/20
it′s the tangent in the inflection point which also intersects the curve in the inflection point I once learned these things with these elementar functions y=ax+b, a≠0 y′=a ⇒ constant slope, no curvature y=ax^2 +bx+c, a≠0 y′=2ax+b zero at x=−(b/(2a)) ⇒ extreme point y′′=2a ⇒ constant curvature depending on the sign of a {: ((a<0 ⇒ maximum)),((a>0 ⇒ minimum)) } at x=−(b/(2a)) y=ax^3 +bx^2 +cx+d, a≠0 y′=3ax^2 +2bx+c zeros at x=−((b±(√(b^2 −3ac)))/(3a)) (1) 2 distinct zeros ∈R ⇒ extreme points (2) 1 double zero ∈R ⇒ no extreme points but an inflection point with horizontal tangent (3) 2 zeros ∉R ⇒ no extreme points in both cases y′′=6ax+2b ⇒ curvature changes zero at x=−(b/(3a)) ⇒ inflection point y′′′=6a { ((a<0 ⇒ curvature changes − to +)),((a>0 ⇒ curvature changes + to −)) :} the inflection tangent also intersects the curve at x=−(b/(3a)) y=ax^4 +bx^3 +cx^2 +dx+e, a≠0 y′=4ax^3 +3bx^2 +2cx+d now it′s getting complicated (1) 3 distinct zeros ∈R (2) 2 double and 1 solitaire zeros ∈R (3) one triple zero ∈R (4) 1 zero ∈R and 2 zeros ∉R we can get 3, 2 or extremes, flat points and a saddle point saddle point: y=x^2 , y′=2x, y′′=2>0 minimum at y=0 y=x^3 , y′=3x^2 , y′′=6x, y′′′=6>0 no extremes inflection point at x=0 with horizontal tangent, curvature changes from − to + y=x^4 , y′=4x^3 , y′′=12x^2 , y′′′=24x minimum at x=0 but y′′=0!? ⇒^? ⇒^? inflection point? but the curvature doesn′t change (y′′′=0) ⇒ this is called a saddle point
$$\mathrm{it}'\mathrm{s}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{in}\:\mathrm{the}\:\mathrm{inflection}\:\mathrm{point}\:\mathrm{which} \\ $$$$\mathrm{also}\:\mathrm{intersects}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{in}\:\mathrm{the}\:\mathrm{inflection} \\ $$$$\mathrm{point} \\ $$$$\mathrm{I}\:\mathrm{once}\:\mathrm{learned}\:\mathrm{these}\:\mathrm{things}\:\mathrm{with}\:\mathrm{these} \\ $$$$\mathrm{elementar}\:\mathrm{functions} \\ $$$$ \\ $$$${y}={ax}+{b},\:{a}\neq\mathrm{0} \\ $$$${y}'={a}\:\Rightarrow\:\mathrm{constant}\:\mathrm{slope},\:\mathrm{no}\:\mathrm{curvature} \\ $$$$ \\ $$$${y}={ax}^{\mathrm{2}} +{bx}+{c},\:{a}\neq\mathrm{0} \\ $$$${y}'=\mathrm{2}{ax}+{b} \\ $$$$\:\:\:\:\:\mathrm{zero}\:\mathrm{at}\:{x}=−\frac{{b}}{\mathrm{2}{a}} \\ $$$$\:\:\:\:\:\Rightarrow\:\mathrm{extreme}\:\mathrm{point} \\ $$$${y}''=\mathrm{2}{a}\:\Rightarrow\:\mathrm{constant}\:\mathrm{curvature}\:\mathrm{depending}\:\mathrm{on} \\ $$$$\:\:\:\:\:\mathrm{the}\:\mathrm{sign}\:\mathrm{of}\:{a} \\ $$$$\:\:\:\:\:\left.\begin{matrix}{{a}<\mathrm{0}\:\Rightarrow\:\mathrm{maximum}}\\{{a}>\mathrm{0}\:\Rightarrow\:\mathrm{minimum}}\end{matrix}\right\}\:\mathrm{at}\:{x}=−\frac{{b}}{\mathrm{2}{a}} \\ $$$$ \\ $$$${y}={ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d},\:{a}\neq\mathrm{0} \\ $$$${y}'=\mathrm{3}{ax}^{\mathrm{2}} +\mathrm{2}{bx}+{c} \\ $$$$\:\:\:\:\:\mathrm{zeros}\:\mathrm{at}\:{x}=−\frac{{b}\pm\sqrt{{b}^{\mathrm{2}} −\mathrm{3}{ac}}}{\mathrm{3}{a}} \\ $$$$\:\:\:\:\:\left(\mathrm{1}\right)\:\mathrm{2}\:\mathrm{distinct}\:\mathrm{zeros}\:\in\mathbb{R}\:\Rightarrow\:\mathrm{extreme}\:\mathrm{points} \\ $$$$\:\:\:\:\:\left(\mathrm{2}\right)\:\mathrm{1}\:\mathrm{double}\:\mathrm{zero}\:\in\mathbb{R}\:\Rightarrow\:\mathrm{no}\:\mathrm{extreme}\:\mathrm{points} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{but}\:\mathrm{an}\:\mathrm{inflection}\:\mathrm{point}\:\mathrm{with}\:\mathrm{horizontal} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{tangent} \\ $$$$\:\:\:\:\:\left(\mathrm{3}\right)\:\mathrm{2}\:\mathrm{zeros}\:\notin\mathbb{R}\:\Rightarrow\:\mathrm{no}\:\mathrm{extreme}\:\mathrm{points} \\ $$$$\mathrm{in}\:\mathrm{both}\:\mathrm{cases} \\ $$$${y}''=\mathrm{6}{ax}+\mathrm{2}{b}\:\Rightarrow\:\mathrm{curvature}\:\mathrm{changes} \\ $$$$\:\:\:\:\:\mathrm{zero}\:\mathrm{at}\:{x}=−\frac{{b}}{\mathrm{3}{a}} \\ $$$$\:\:\:\:\:\Rightarrow\:\mathrm{inflection}\:\mathrm{point} \\ $$$${y}'''=\mathrm{6}{a}\:\begin{cases}{{a}<\mathrm{0}\:\Rightarrow\:\mathrm{curvature}\:\mathrm{changes}\:−\:\mathrm{to}\:+}\\{{a}>\mathrm{0}\:\Rightarrow\:\mathrm{curvature}\:\mathrm{changes}\:+\:\mathrm{to}\:−}\end{cases} \\ $$$$\mathrm{the}\:\mathrm{inflection}\:\mathrm{tangent}\:\mathrm{also}\:\mathrm{intersects}\:\mathrm{the} \\ $$$$\mathrm{curve}\:\mathrm{at}\:{x}=−\frac{{b}}{\mathrm{3}{a}} \\ $$$$ \\ $$$${y}={ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+{e},\:{a}\neq\mathrm{0} \\ $$$${y}'=\mathrm{4}{ax}^{\mathrm{3}} +\mathrm{3}{bx}^{\mathrm{2}} +\mathrm{2}{cx}+{d} \\ $$$$\mathrm{now}\:\mathrm{it}'\mathrm{s}\:\mathrm{getting}\:\mathrm{complicated} \\ $$$$\:\:\:\:\:\left(\mathrm{1}\right)\:\mathrm{3}\:\mathrm{distinct}\:\mathrm{zeros}\:\in\mathbb{R} \\ $$$$\:\:\:\:\:\left(\mathrm{2}\right)\:\mathrm{2}\:\mathrm{double}\:\mathrm{and}\:\mathrm{1}\:\mathrm{solitaire}\:\mathrm{zeros}\:\in\mathbb{R} \\ $$$$\:\:\:\:\:\left(\mathrm{3}\right)\:\mathrm{one}\:\mathrm{triple}\:\mathrm{zero}\:\in\mathbb{R} \\ $$$$\:\:\:\:\:\left(\mathrm{4}\right)\:\mathrm{1}\:\mathrm{zero}\:\in\mathbb{R}\:\mathrm{and}\:\mathrm{2}\:\mathrm{zeros}\:\notin\mathbb{R} \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{get}\:\mathrm{3},\:\mathrm{2}\:\mathrm{or}\:\mathrm{extremes},\:\mathrm{flat}\:\mathrm{points}\:\mathrm{and} \\ $$$$\mathrm{a}\:\mathrm{saddle}\:\mathrm{point} \\ $$$$\mathrm{saddle}\:\mathrm{point}: \\ $$$${y}={x}^{\mathrm{2}} ,\:{y}'=\mathrm{2}{x},\:{y}''=\mathrm{2}>\mathrm{0} \\ $$$$\:\:\:\:\:\mathrm{minimum}\:\mathrm{at}\:{y}=\mathrm{0} \\ $$$${y}={x}^{\mathrm{3}} ,\:{y}'=\mathrm{3}{x}^{\mathrm{2}} ,\:{y}''=\mathrm{6}{x},\:{y}'''=\mathrm{6}>\mathrm{0} \\ $$$$\:\:\:\:\:\mathrm{no}\:\mathrm{extremes} \\ $$$$\:\:\:\:\:\mathrm{inflection}\:\mathrm{point}\:\mathrm{at}\:{x}=\mathrm{0}\:\mathrm{with}\:\mathrm{horizontal} \\ $$$$\:\:\:\:\:\mathrm{tangent},\:\mathrm{curvature}\:\mathrm{changes}\:\mathrm{from}\:−\:\mathrm{to}\:+ \\ $$$${y}={x}^{\mathrm{4}} ,\:{y}'=\mathrm{4}{x}^{\mathrm{3}} ,\:{y}''=\mathrm{12}{x}^{\mathrm{2}} ,\:{y}'''=\mathrm{24}{x} \\ $$$$\:\:\:\:\:\mathrm{minimum}\:\mathrm{at}\:{x}=\mathrm{0}\:\mathrm{but}\:{y}''=\mathrm{0}!?\:\overset{?} {\Rightarrow} \\ $$$$\:\:\:\:\:\overset{?} {\Rightarrow}\:\mathrm{inflection}\:\mathrm{point}?\:\mathrm{but}\:\mathrm{the}\:\mathrm{curvature} \\ $$$$\:\:\:\:\:\mathrm{doesn}'\mathrm{t}\:\mathrm{change}\:\left({y}'''=\mathrm{0}\right)\:\Rightarrow\:\mathrm{this}\:\mathrm{is}\:\mathrm{called} \\ $$$$\:\:\:\:\:\mathrm{a}\:\mathrm{saddle}\:\mathrm{point} \\ $$
Commented by  M±th+et+s last updated on 01/May/20
very cool explanation.thank you very much
$${very}\:{cool}\:{explanation}.{thank}\:{you}\:{very}\:{much} \\ $$

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