Horizontal and non Horizontal Points of Inlexion

A stationary point  
\[(x_0, f(x_0))\]
  is a point of inflexion on a curve  
\[f(x)\]
  if  
\[\frac{d^2f}{dx^2} \|_{x=x_0}=0\]
.
If also  
\[\frac{df}{dx} \|_{x=x_0}=0\]
  the point is a horizontal point of inflexion.
Example:  
\[f(x)=x^3(x-1)=x^4-x^3\]
.
\[\frac{df}{dx}=4x^3-3x^2\]
  so  
\[\frac{df}{dx}=0\]
  at  
\[x=0\]
\[\frac{d^f}{dx^2}=12x^2-6x\]
  so  
\[\frac{d^2f}{dx^2}=0\]
  at  
\[x=0\]
.
The point  
\[(0,f(0))=(0,0)\]
  is a stationary point of inflexion.

point of inflexion

Eample:  
\[f(x)=x(x^2+2)=x^3+2x\]
.
\[\frac{df}{dx}=3x^2+2, \: \frac{d^2f}{dx^2}=6x\]
.
When  
\[x=0\]
,  
\[\frac{df}{dx} \neq 0, \: \frac{d^2f}{dx^2} = 0\]
.
When  
\[x=0\]
, the point is a point of inflexion but not a stationary point of inflexion.

non stationary point of inflexion

Add comment

Security code
Refresh