What is the inflection (2-x^2)/(1+x^4) ?

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

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inflection (2-x^2)/(1+x^4)

inflection points

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

(- \frac{7.14711 E 15}{1.0 E 15}, - \frac{5.14711 E 30}{1.0 E 1 5 ^{2} + 7.14711 E 1 5 ^{2}}), (- \frac{2.98799 E 15}{5.0 E 15}, \frac{3.50601 E 31}{5.0 E 1 5 ^{2} + 2.98799 E 1 5 ^{2}}), (\frac{2.98799 E 15}{5.0 E 15}, \frac{3.50601 E 31}{5.0 E 1 5 ^{2} + 2.98799 E 1 5 ^{2}}), (\frac{7.14711 E 15}{1.0 E 15}, - \frac{5.14711 E 30}{1.0 E 1 5 ^{2} + 7.14711 E 1 5 ^{2}})

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x = - \frac{7.14711 E 15}{1.0 E 15}, x = - \frac{2.98799 E 15}{5.0 E 15}, x = \frac{2.98799 E 15}{5.0 E 15}, x = \frac{7.14711 E 15}{1.0 E 15}

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

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

All inflection points are in domain and not on domain edges

x = - \frac{7.14711 E 15}{1.0 E 15}, x = - \frac{2.98799 E 15}{5.0 E 15}, x = \frac{2.98799 E 15}{5.0 E 15}, x = \frac{7.14711 E 15}{1.0 E 15}

Find intervals:

Concave Downward

: - \infty < x < - \frac{7.14711 E 15}{1.0 E 15},

Concave Upward

: - \frac{7.14711 E 15}{1.0 E 15} < x < - \frac{2.98799 E 15}{5.0 E 15},

Concave Downward

: - \frac{2.98799 E 15}{5.0 E 15} < x < \frac{2.98799 E 15}{5.0 E 15},

Concave Upward

: \frac{2.98799 E 15}{5.0 E 15} < x < \frac{7.14711 E 15}{1.0 E 15},

Concave Downward

: \frac{7.14711 E 15}{1.0 E 15} < x < \infty

Plug

x = - \frac{7.14711 E 15}{1.0 E 15}

into

\frac{2 - x ^{2}}{1 + x ^{4}} : - \frac{5.14711 E 30}{1.0 E 1 5 ^{2} + 7.14711 E 1 5 ^{2}}

(- \frac{7.14711 E 15}{1.0 E 15}, - \frac{5.14711 E 30}{1.0 E 1 5 ^{2} + 7.14711 E 1 5 ^{2}})

Plug

x = - \frac{2.98799 E 15}{5.0 E 15}

into

\frac{2 - x ^{2}}{1 + x ^{4}} : \frac{3.50601 E 31}{5.0 E 1 5 ^{2} + 2.98799 E 1 5 ^{2}}

(- \frac{2.98799 E 15}{5.0 E 15}, \frac{3.50601 E 31}{5.0 E 1 5 ^{2} + 2.98799 E 1 5 ^{2}})

Plug

x = \frac{2.98799 E 15}{5.0 E 15}

into

\frac{2 - x ^{2}}{1 + x ^{4}} : \frac{3.50601 E 31}{5.0 E 1 5 ^{2} + 2.98799 E 1 5 ^{2}}

(\frac{2.98799 E 15}{5.0 E 15}, \frac{3.50601 E 31}{5.0 E 1 5 ^{2} + 2.98799 E 1 5 ^{2}})

Plug

x = \frac{7.14711 E 15}{1.0 E 15}

into

\frac{2 - x ^{2}}{1 + x ^{4}} : - \frac{5.14711 E 30}{1.0 E 1 5 ^{2} + 7.14711 E 1 5 ^{2}}

(\frac{7.14711 E 15}{1.0 E 15}, - \frac{5.14711 E 30}{1.0 E 1 5 ^{2} + 7.14711 E 1 5 ^{2}})

(- \frac{7.14711 E 15}{1.0 E 15}, - \frac{5.14711 E 30}{1.0 E 1 5 ^{2} + 7.14711 E 1 5 ^{2}}), (- \frac{2.98799 E 15}{5.0 E 15}, \frac{3.50601 E 31}{5.0 E 1 5 ^{2} + 2.98799 E 1 5 ^{2}}), (\frac{2.98799 E 15}{5.0 E 15}, \frac{3.50601 E 31}{5.0 E 1 5 ^{2} + 2.98799 E 1 5 ^{2}}), (\frac{7.14711 E 15}{1.0 E 15}, - \frac{5.14711 E 30}{1.0 E 1 5 ^{2} + 7.14711 E 1 5 ^{2}})

f (x) = \frac{5 x - 10}{x ^{2} + x - 12}

f (x) = (\frac{7}{x}) + (\frac{1}{x ^{2}})

f (x) = \frac{5}{- x ^{2} - 3 x + 4}

(- 6, 6), (2, - 4)

y = x + 7