Inflection points

Summary

• If \(x_0\) is a strict local interior extreme point of \(f'\) then \(x_0\) is called an inflection point of \(f\) .
• If \(x_0\) is an inflection point of \(f\) and \(f\) is twice differentiable at \(x_0\) then \(f^{''}\left( x \right)=0\) (the opposite is not necessarily true).
• Examples:
• If \(f\left( x \right)=x^3\) then \(f'\left( x \right)=3x^2\) . Since \(x_0=0\) is a strict local minimum point of \(f'\) , \(x_0=0\) is an inflection point of \(f\) . \(f^{''}\left( x \right)=6x\) and \(f^{''}\left( 0 \right)=0\) as expected.
• If \(f\left( x \right)=x^4\) then \(f'\left( x \right)=4x^3\) and \(f^{''}\left( x \right)=12x^2\) . Even though \(f^{''}\left( 0 \right)=0\) , \(x_0=0\) is not an inflection point of \(f\) as \(x_0=0\) is not a local extreme point of \(f'\) (it is a saddle point).
• If \(x_0\) is an inflection point of \(f\) then \(f\) is strictly convex just to the left of \(x_0\) and strictly concave just to the right of \(x_0\) or vice versa.
• Finding inflections points of \(f\) is the same as finding local extreme points of \(f'\) excluding possible boundary points:
• Find all \(x\) -values where \(f^{''}\left( x \right)=0\) .
• Check that they are local extreme points, either by checking that \(f^{'''}\left( x_c \right)≠0\) or by checking that \(f^{''}\left( x_c \right)\) changes sign at \(x=x_0\) ( \(x_c\) is a candidate, \(f^{''}\left( x_c \right)=0\) ).