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

The inflection f(x)=-3x^5+5x^3 is (-1/(sqrt(2)),-7/(4sqrt(2))),(0,0),(1/(sqrt(2)), 7/(4sqrt(2)))

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inflection f(x)=-3x^5+5x^3

inflection points

f (x) = - 3 x^{5} + 5 x^{3}

(- \frac{1}{2}, - \frac{7}{4 2}), (0, 0), (\frac{1}{2}, \frac{7}{4 2})

Solution steps

f^{''} (x) = - 60 x^{3} + 30 x

Find intervals:

Concave Upward

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

Concave Downward

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

Concave Upward

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

Concave Downward

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

Plug

x = - \frac{1}{2}

into

- 3 x^{5} + 5 x^{3} : - \frac{7}{4 2}

(- \frac{1}{2}, - \frac{7}{4 2})

Plug

x = 0

into

- 3 x^{5} + 5 x^{3} : 0

(0, 0)

Plug

x = \frac{1}{2}

into

- 3 x^{5} + 5 x^{3} : \frac{7}{4 2}

(\frac{1}{2}, \frac{7}{4 2})

(- \frac{1}{2}, - \frac{7}{4 2}), (0, 0), (\frac{1}{2}, \frac{7}{4 2})

What is the inflection f(x)=-3x^5+5x^3 ?
The inflection f(x)=-3x^5+5x^3 is (-1/(sqrt(2)),-7/(4sqrt(2))),(0,0),(1/(sqrt(2)), 7/(4sqrt(2)))