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

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

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

inflection points

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

(- 2 - 3, \frac{1 - 3}{4}), (- 2 + 3, \frac{3 + 1}{4}), (1, 1)

Solution steps

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

Find intervals:

Concave Downward

: - \infty < x < - 2 - 3,

Concave Upward

: - 2 - 3 < x < - 2 + 3,

Concave Downward

: - 2 + 3 < x < 1,

Concave Upward

: 1 < x < \infty

Plug

x = - 2 - 3

into

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

(- 2 - 3, \frac{1 - 3}{4})

Plug

x = - 2 + 3

into

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

(- 2 + 3, \frac{3 + 1}{4})

Plug

x = 1

into

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

(1, 1)

(- 2 - 3, \frac{1 - 3}{4}), (- 2 + 3, \frac{3 + 1}{4}), (1, 1)

y \leq 1

f (x) = x^{3} + 6

\frac{x ^{2} + 2 x + 3}{x ^{3} + x}

f (x) = x^{2} - 10 x + 21

f (x) = \frac{x + 9}{x ^{2} + 5}

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