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

The inflection (x^2+x+1)/(x^2-x+1) is (-1.87938…,0.41374…),(0.34729…,1.89819…),(1.53208…,2.68805…)

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

inflection points

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

(- 1.87938 \dots, 0.41374 \dots), (0.34729 \dots, 1.89819 \dots), (1.53208 \dots, 2.68805 \dots)

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x = - 1.87938 \dots, x \approx 0.34729 \dots, x \approx 1.53208 \dots

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

\frac{x ^{2} + x + 1}{x ^{2} - x + 1} : - \infty < x < \infty

All inflection points are in domain and not on domain edges

x = - 1.87938 \dots, x \approx 0.34729 \dots, x \approx 1.53208 \dots

Find intervals:

Concave Downward

: - \infty < x < - 1.87938 \dots,

Concave Upward

: - 1.87938 \dots < x < 0.34729 \dots,

Concave Downward

: 0.34729 \dots < x < 1.53208 \dots,

Concave Upward

: 1.53208 \dots < x < \infty

Plug

x = - 1.87938 \dots

into

\frac{x ^{2} + x + 1}{x ^{2} - x + 1} : 0.41374 \dots

(- 1.87938 \dots, 0.41374 \dots)

Plug

x = 0.34729 \dots

into

\frac{x ^{2} + x + 1}{x ^{2} - x + 1} : 1.89819 \dots

(0.34729 \dots, 1.89819 \dots)

Plug

x = 1.53208 \dots

into

\frac{x ^{2} + x + 1}{x ^{2} - x + 1} : 2.68805 \dots

(1.53208 \dots, 2.68805 \dots)

(- 1.87938 \dots, 0.41374 \dots), (0.34729 \dots, 1.89819 \dots), (1.53208 \dots, 2.68805 \dots)

f (x) = (x - 4)^{2}

\frac{1}{x ^{2} - 10 x + 15}

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

f (x) = 2 x + 3 - 1

f (x) = 3 + 7 x

What is the inflection (x^2+x+1)/(x^2-x+1) ?
The inflection (x^2+x+1)/(x^2-x+1) is (-1.87938…,0.41374…),(0.34729…,1.89819…),(1.53208…,2.68805…)