What is the inflection f(x)= x/(x^2+1) ?

The inflection f(x)= x/(x^2+1) is (-sqrt(3),-(sqrt(3))/4),(0,0),(sqrt(3),(sqrt(3))/4)

inflection f(x)= x/(x^2+1)

inflection points

f (x) = \frac{x}{x ^{2} + 1}

(- 3, - \frac{3}{4}), (0, 0), (3, \frac{3}{4})

Solution steps

f^{''} (x) = - \frac{2 x ( - x ^{2} + 3 )}{( x ^{2} + 1 ) ^{3}}

Find intervals:

Concave Downward

: - \infty < x < - 3,

Concave Upward

: - 3 < x < 0,

Concave Downward

: 0 < x < 3,

Concave Upward

: 3 < x < \infty

Plug

x = - 3

into

\frac{x}{x ^{2} + 1} : - \frac{3}{4}

(- 3, - \frac{3}{4})

Plug

x = 0

into

\frac{x}{x ^{2} + 1} : 0

(0, 0)

Plug

x = 3

into

\frac{x}{x ^{2} + 1} : \frac{3}{4}

(3, \frac{3}{4})

(- 3, - \frac{3}{4}), (0, 0), (3, \frac{3}{4})

e x t re m e f (x) = 2 x 2 x^{2} + 4

sy mm e t ry x^{2} + 6 x + 3

v er t i ces y = x^{2} - 6 x + 18

in v erse (x + 6)^{5}

in v erse f (x) = (x + 3)^{\frac{1}{3}} - 4

What is the inflection f(x)= x/(x^2+1) ?
The inflection f(x)= x/(x^2+1) is (-sqrt(3),-(sqrt(3))/4),(0,0),(sqrt(3),(sqrt(3))/4)