What is the inflection f(x)=(3x^2)/(5+x^2) ?

The inflection f(x)=(3x^2)/(5+x^2) is (-sqrt(5/3), 3/4),(sqrt(5/3), 3/4)

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inflection f(x)=(3x^2)/(5+x^2)

inflection points

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

(- \frac{5}{3}, \frac{3}{4}), (\frac{5}{3}, \frac{3}{4})

Solution steps

f^{''} (x) = \frac{30 ( - 3 x ^{2} + 5 )}{( 5 + x ^{2} ) ^{3}}

Find intervals:

Concave Downward

: - \infty < x < - \frac{5}{3},

Concave Upward

: - \frac{5}{3} < x < \frac{5}{3},

Concave Downward

: \frac{5}{3} < x < \infty

Plug

x = - \frac{5}{3}

into

\frac{3 x ^{2}}{5 + x ^{2}} : \frac{3}{4}

(- \frac{5}{3}, \frac{3}{4})

Plug

x = \frac{5}{3}

into

\frac{3 x ^{2}}{5 + x ^{2}} : \frac{3}{4}

(\frac{5}{3}, \frac{3}{4})

(- \frac{5}{3}, \frac{3}{4}), (\frac{5}{3}, \frac{3}{4})

f (x) = \frac{4 x ^{2} + 1}{2 x}

(3, 3), (2, - 2)

f (x) = 3 x - ln (3 x)

\frac{x ^{2} - 16}{x ^{2} - 1}

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

What is the inflection f(x)=(3x^2)/(5+x^2) ?
The inflection f(x)=(3x^2)/(5+x^2) is (-sqrt(5/3), 3/4),(sqrt(5/3), 3/4)