To find the inflection points the first step we have to follow is to find the second and third derivatives of the function:
[tex]\begin{gathered} f\mleft(x\mright)=x^3-9x^2+10 \\ f^{\prime}\left(x\right)=3x^2-18x \\ f^{\prime}^{\prime}\left(x\right)=6x-18 \\ f^{\prime}^{\prime}^{\prime}\left(x\right)=6 \end{gathered}[/tex]Now, find the values of x for which the second derivative is 0:
[tex]\begin{gathered} 0=6x-18 \\ 18=6x \\ x=\frac{18}{6} \\ x=3 \end{gathered}[/tex]Evaluate the third derivative at this values of x, if the third derivative is different from 0, then that value is an inflection point:
[tex]f^{\prime}^{\prime}^{\prime}\left(3\right)=6[/tex]It means that there is an inflection point at x=3.
