What is the inflection xe^{(-x^2)/2} ?

The inflection xe^{(-x^2)/2} is (-3^{1/2},-(3/(e^3))^{1/2}),(0,0),(3^{1/2},(3/(e^3))^{1/2})

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inflection xe^{(-x^2)/2}

inflection points

x e^{\frac{- x ^{2}}{2}}

(- 3^{\frac{1}{2}}, - (\frac{3}{e ^{3}})^{\frac{1}{2}}), (0, 0), (3^{\frac{1}{2}}, (\frac{3}{e ^{3}})^{\frac{1}{2}})

Solution steps

Find where

f^{''} (x)

is equal to zero or undefined

x = - 3^{\frac{1}{2}}, x = 0, x = 3^{\frac{1}{2}}

Identify inflection points not in domain or where

f (x)

is not continuous

Domain of

x e^{\frac{- x ^{2}}{2}} : - \infty < x < \infty

All inflection points are in domain and not on domain edges

x = - 3^{\frac{1}{2}}, x = 0, x = 3^{\frac{1}{2}}

Find intervals:

Concave Downward

: - \infty < x < - 3^{\frac{1}{2}},

Concave Upward

: - 3^{\frac{1}{2}} < x < 0,

Concave Downward

: 0 < x < 3^{\frac{1}{2}},

Concave Upward

: 3^{\frac{1}{2}} < x < \infty

Plug

x = - 3^{\frac{1}{2}}

into

x e^{\frac{- x ^{2}}{2}} : - (\frac{3}{e ^{3}})^{\frac{1}{2}}

(- 3^{\frac{1}{2}}, - (\frac{3}{e ^{3}})^{\frac{1}{2}})

Plug

x = 0

into

x e^{\frac{- x ^{2}}{2}} : 0

(0, 0)

Plug

x = 3^{\frac{1}{2}}

into

x e^{\frac{- x ^{2}}{2}} : (\frac{3}{e ^{3}})^{\frac{1}{2}}

(3^{\frac{1}{2}}, (\frac{3}{e ^{3}})^{\frac{1}{2}})

(- 3^{\frac{1}{2}}, - (\frac{3}{e ^{3}})^{\frac{1}{2}}), (0, 0), (3^{\frac{1}{2}}, (\frac{3}{e ^{3}})^{\frac{1}{2}})

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

(2, - 3), (10, 7)

f (x) = - 7 + 6 x - x^{3}

f (x) = \frac{2}{6 - 5 x}

f (x) = x^{3} - 3

What is the inflection xe^{(-x^2)/2} ?
The inflection xe^{(-x^2)/2} is (-3^{1/2},-(3/(e^3))^{1/2}),(0,0),(3^{1/2},(3/(e^3))^{1/2})