Respuesta :
Given
[tex]f(x)=x^4-5x^3[/tex]To find its inflection points, solve the equation f''(x)=0 for x, as shown below
[tex]\begin{gathered} \Rightarrow f^{\prime}(x)=4x^3-5*3x^2=4x^3-15x^2 \\ \Rightarrow f^{\prime}(x)=4x^3-15x^2 \\ \\ \end{gathered}[/tex]Finding the second derivative,
[tex]\begin{gathered} \Rightarrow f^{\prime}^{\prime}(x)=12x^2-30x \\ \Rightarrow f^{\prime}^{\prime}(x)=0 \\ \Rightarrow12x^2-30x=0 \\ \Rightarrow x(12x-30)=0 \\ \Rightarrow x=0,x=\frac{30}{12}=\frac{5}{2} \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} f(0)=0 \\ f(\frac{5}{2})=(\frac{5}{2})^4-5(\frac{5}{2})^3=-\frac{625}{16} \end{gathered}[/tex]The two inflection points are
[tex]\begin{gathered} (0,0) \\ and \\ (\frac{5}{2},-\frac{625}{16}) \end{gathered}[/tex]Do not confuse inflection points with critical points.
