\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&15x^{2} - 20x - 10\\&\text{And our second is:}\\&30x - 20\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=0.667\\&\text{We find one real unique root at the point }x=0.667\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&7 - 6x^{2} - 4x\\&\text{And our second is:}\\&- 12x - 4\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.333\\&\text{We find one real unique root at the point }x=-0.333\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&9x^{2} - 18x + 32x^{3} + 40x^{4} + 8\\&\text{And our second is:}\\&18x + 96x^{2} + 160x^{3} - 18\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=0.297,-0.448+0.422i,-0.448-0.422i\\&\text{We find one real unique root at the point }x=0.297\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&6x^{2} - 6x - 8x^{3} - 40x^{4}\\&\text{And our second is:}\\&12x - 24x^{2} - 160x^{3} - 6\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.475,0.162-0.230i,0.162+0.230i\\&\text{We find one real unique root at the point }x=-0.475\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&16x - 9x^{2} - 40x^{3} + 3\\&\text{And our second is:}\\&16 - 120x^{2} - 18x\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.448,0.298\\&\text{We find real unique roots at the points }x=-0.448,0.298\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&3x^{2} -\frac{ (3ln(2))}{2^{(3x)}}- 20x - 40x^{3} + 30x^{4} - 1\\&\text{And our second is:}\\&6x +\frac{ (9ln(2)^{2})}{2^{(3x)}}- 120x^{2} + 120x^{3} - 20\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=1.091\\&\text{We find one real unique root at the point }x=1.091\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&3\cdot 3^{(3x)}ln(3) + 30x^{2} + 40x^{3} + 10x^{4} - 4\\&\text{And our second is:}\\&60x + 9\cdot 3^{(3x)}ln(3)^{2} + 120x^{2} + 40x^{3}\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.154\\&\text{We find one real unique root at the point }x=-0.154\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&16x + 2e^{(2x)} + 15x^{2} - 8x^{3} - 35x^{4} - 8\\&\text{And our second is:}\\&30x + 4e^{(2x)} - 24x^{2} - 140x^{3} + 16\\&\text{For a reminder of derivative rules:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=0.660\\&\text{We find one real unique root at the point }x=0.660\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&\frac{1}{x}- 12x - 30x^{2} + 24x^{3} + 4\\&\text{And our second is:}\\&72x^{2} -\frac{ 1}{x^{2}}- 60x - 12\\&\text{For a reminder of derivative rules:}\end{align*}\)

\(\displaystyle \begin{align*}\\&d[ln(ax)]=\frac{dx}{x}\\&d[x^a]=ax^{a-1}dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=-0.292,1.012,0.057-0.209i,0.057+0.209i\\&\text{We find real unique roots at the points }x=-0.292,1.012\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative. For the}\\&\text{function we're given, our first derivative is :}\\&16x +\frac{ 1}{x}- 30x^{2} + 20x^{3}\\&\text{And our second is:}\\&60x^{2} -\frac{ 1}{x^{2}}- 60x + 16\\&\text{For a reminder of derivative rules:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[x^a]=ax^{a-1}dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=-0.185,0.651,0.267-0.259i,0.267+0.259i\\&\text{We find real unique roots at the points }x=-0.185,0.651\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}\)