Determine the points of inflexion of a function Calculus

Determine the x-coordinates of the points of inflexion of the following function:

f\left(x\right)=5x^{3}-30x^{2}+25x-10

x=-2

x=2

x=-1

x=1

Determine the x-coordinates of the points of inflexion of the following function:

f\left(x\right)=x^{2}e^{3x}

x=\dfrac{-2-\sqrt{2}}{3} and x=\dfrac{-2+\sqrt{2}}{3}

x=\dfrac{-3-2\sqrt{3}}{2} and x=\dfrac{-3+2\sqrt{3}}{2}

x=-1 and x=2

There are no inflexion points.

Determine the x-coordinates of the points of inflexion of the following function:

f\left(x\right)=x^{3}-5x^{2}+7x-3

x=-\dfrac{7}{3} and x=-1

x=1 and x=\dfrac{7}{3}

x=\dfrac{5}{3}

There are no inflexion points.

Determine the x-coordinates of the points of inflexion of the following function:

f\left(x\right)=\ln\left(x^{2}+4x\right)

x=-4 and x=2

x=-\dfrac{1}{2} and x=1

x=-2

There are no inflexion points.

Determine the x-coordinates of the points of inflexion of the following function:

f\left(x\right)=\dfrac{1}{2}x^{4}+\dfrac{2}{3}x^{3}-5x+4

x=0 and x=-\dfrac{2}{3}

x=-\dfrac{1}{2} and x=1

x=-\dfrac{3}{4} and x=\dfrac{1}{3}

There are no inflexion points.

Determine the x-coordinates of the points of inflexion of the following function:

f\left(x\right)=x^{3}\ln\left(x\right)

x={e^{-\frac{5}{6}}}

x=e^{3}

x=3e^{2}

There are no inflexion points.

Determine the x-coordinates of the points of inflexion of the following function:

f\left(x\right)=4x^{2}-10x+\ln\left(7\right)

x=\dfrac{5}{4}

x=0

x=e^{7}

There are no inflexion points.