What is the inflection x/(x^2+2x+3) ?

The inflection x/(x^2+2x+3) is (-2.58422…,-0.57302…),(-0.70571…,-0.33821…),(3.28994…,0.16124…)

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inflection x/(x^2+2x+3)

inflection points

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

(- 2.58422 \dots, - 0.57302 \dots), (- 0.70571 \dots, - 0.33821 \dots), (3.28994 \dots, 0.16124 \dots)

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x \approx - 2.58422 \dots, x \approx - 0.70571 \dots, x = 3.28994 \dots

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

\frac{x}{x ^{2} + 2 x + 3} : - \infty < x < \infty

All inflection points are in domain and not on domain edges

x \approx - 2.58422 \dots, x \approx - 0.70571 \dots, x = 3.28994 \dots

Find intervals:

Concave Downward

: - \infty < x < - 2.58422 \dots,

Concave Upward

: - 2.58422 \dots < x < - 0.70571 \dots,

Concave Downward

: - 0.70571 \dots < x < 3.28994 \dots,

Concave Upward

: 3.28994 \dots < x < \infty

Plug

x = - 2.58422 \dots

into

\frac{x}{x ^{2} + 2 x + 3} : - 0.57302 \dots

(- 2.58422 \dots, - 0.57302 \dots)

Plug

x = - 0.70571 \dots

into

\frac{x}{x ^{2} + 2 x + 3} : - 0.33821 \dots

(- 0.70571 \dots, - 0.33821 \dots)

Plug

x = 3.28994 \dots

into

\frac{x}{x ^{2} + 2 x + 3} : 0.16124 \dots

(3.28994 \dots, 0.16124 \dots)

(- 2.58422 \dots, - 0.57302 \dots), (- 0.70571 \dots, - 0.33821 \dots), (3.28994 \dots, 0.16124 \dots)

f (x) = 10 - 3 - x

(x + 3)

f (x) = - x^{2} - 6 x

6 x - 4 y = - 6

y = \frac{3}{2} x - \frac{13}{2}

What is the inflection x/(x^2+2x+3) ?
The inflection x/(x^2+2x+3) is (-2.58422…,-0.57302…),(-0.70571…,-0.33821…),(3.28994…,0.16124…)