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

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

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

inflection points

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

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

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

Plug

x = 0

into

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

(0, 0)

Plug

x = 3

into

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

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

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

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