Question

For the function f(x) = x^4 - 5x^3, find inflection points A) Inflection point(s) of f(x):

Answer 1

Given

`f(x)=x^4-5x^3`

To find its inflection points, solve the equation f''(x)=0 for x, as shown below

`\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}`

Finding the second derivative,

`\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=(30)/(12)=(5)/(2) \end{gathered}`

Therefore,

`\begin{gathered} f(0)=0 \\ f((5)/(2))=((5)/(2))^4-5((5)/(2))^3=-(625)/(16) \end{gathered}`

The two inflection points are

`\begin{gathered} (0,0) \\ and \\ ((5)/(2),-(625)/(16)) \end{gathered}`

Do not confuse inflection points with critical points.