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, $f^{''}\left( x_c \right)=0$ ).