What is the inflection (x^3)/(x^2+12) ?

The inflection (x^3)/(x^2+12) is (-6,-9/2),(0,0),(6, 9/2)

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inflection (x^3)/(x^2+12)

inflection points

\frac{x ^{3}}{x ^{2} + 12}

(- 6, - \frac{9}{2}), (0, 0), (6, \frac{9}{2})

Solution steps

f^{''} (x) = \frac{24 x ( - x ^{2} + 36 )}{( x ^{2} + 12 ) ^{3}}

Find intervals:

Concave Upward

: - \infty < x < - 6,

Concave Downward

: - 6 < x < 0,

Concave Upward

: 0 < x < 6,

Concave Downward

: 6 < x < \infty

Plug

x = - 6

into

\frac{x ^{3}}{x ^{2} + 12} : - \frac{9}{2}

(- 6, - \frac{9}{2})

Plug

x = 0

into

\frac{x ^{3}}{x ^{2} + 12} : 0

(0, 0)

Plug

x = 6

into

\frac{x ^{3}}{x ^{2} + 12} : \frac{9}{2}

(6, \frac{9}{2})

(- 6, - \frac{9}{2}), (0, 0), (6, \frac{9}{2})

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

f (x) = \frac{3 x ^{2} - 3 x}{x ^{2} + x - 12}

f (x) = \frac{2 x - 13}{2 x - 6}

(- 4, 5), (4, - 2)

2 y = - 5 x + 5

What is the inflection (x^3)/(x^2+12) ?
The inflection (x^3)/(x^2+12) is (-6,-9/2),(0,0),(6, 9/2)