one-of-the-conditions-of-the-inflection-point-is-inflection-tangent-what-is-inflection-tangent-

Question Number 91460 by M±th+et+s last updated on 30/Apr/20

o n e o f t h e c o n d i t i o n s o f t h e i n f l e c t i o n

p o i n t i s i n f l e c t i o n t a n g e n t .

w h a t i s i n f l e c t i o n t a n g e n t ?

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 = a x + b, a \neq 0

y^{'} = a \Rightarrow constant slope, no curvature

y = {a x}^{2} + b x + c, a \neq 0

y^{'} = 2 a x + b

zero at x = - \frac{b}{2 a}

\Rightarrow extreme point

y^{″} = 2 a \Rightarrow constant curvature depending on

the sign of a

\begin{matrix} a < 0 \Rightarrow maximum \\ a > 0 \Rightarrow minimum \end{matrix}} at x = - \frac{b}{2 a}

y = {a x}^{3} + {b x}^{2} + c x + d, a \neq 0

y^{'} = 3 {a x}^{2} + 2 b x + c

zeros at x = - \frac{b \pm \sqrt{b^{2} - 3 a c}}{3 a}

(1) 2 distinct zeros \in R \Rightarrow extreme points

(2) 1 double zero \in R \Rightarrow no extreme points

but an inflection point with horizontal

tangent

(3) 2 zeros \notin R \Rightarrow no extreme points

in both cases

y^{″} = 6 a x + 2 b \Rightarrow curvature changes

zero at x = - \frac{b}{3 a}

\Rightarrow inflection point

y^{‴} = 6 a {\begin{cases} a < 0 \Rightarrow curvature changes - to + \\ a > 0 \Rightarrow curvature changes + to - \end{cases}

the inflection tangent also intersects the

curve at x = - \frac{b}{3 a}

y = {a x}^{4} + {b x}^{3} + {c x}^{2} + d x + e, a \neq 0

y^{'} = 4 {a x}^{3} + 3 {b x}^{2} + 2 c x + d

now {it}^{'} s getting complicated

(1) 3 distinct zeros \in R

(2) 2 double and 1 solitaire zeros \in R

(3) one triple zero \in R

(4) 1 zero \in R and 2 zeros \notin R

we can get 3, 2 or extremes, flat points and

a saddle point

saddle point :

y = x^{2}, y^{'} = 2 x, y^{″} = 2 > 0

minimum at y = 0

y = x^{3}, y^{'} = 3 x^{2}, y^{″} = 6 x, y^{‴} = 6 > 0

no extremes

inflection point at x = 0 with horizontal

tangent, curvature changes from - to +

y = x^{4}, y^{'} = 4 x^{3}, y^{″} = 12 x^{2}, y^{‴} = 24 x

minimum at x = 0 but y^{″} = 0! ? \overset{?}{\Rightarrow}

\overset{?}{\Rightarrow} inflection point ? but the curvature

{doesn}^{'} t change (y^{‴} = 0) \Rightarrow this is called

a saddle point

Commented by M±th+et+s last updated on 01/May/20

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