#math/calculus
An **inflection point** $(x_{0},f(x_{0}))$ is a point where the graph changes concavity. In practice, locate candidates where $f''(x)=0$ or $f''$ is undefined, then check whether $f''$ actually changes sign across the candidate.
> [!Important]
> The equation $f''(x_{0})=0$ identifies a candidate, not automatically an inflection point. For example, $x^{4}$ has $f''(0)=0$ but no change in concavity. Conversely, $x^{1/3}$ changes concavity at $0$ even though $f''(0)$ is undefined.
If $f''(x_{0})=0$ and $f'''$ is continuous with $f'''(x_{0})\neq0$, then $f''$ changes sign at $x_{0}$, so $x_{0}$ is an inflection point.
![[tikz-2bf96f570305.svg]]
%% tikz-source
```latex
\begin{document}
\begin{tikzpicture}[x=1.15cm,y=1cm,line cap=round]
\draw[->,black!45] (-2.6,0) -- (2.7,0) node[right] {$x$};
\draw[->,black!45] (0,-0.2) -- (0,3.2) node[above] {$y$};
\draw[very thick,blue,domain=-2.25:2.25,samples=120] plot (\x,{0.22*\x^4-1.1*\x^2+2});
\foreach \x in {-0.913,0.913}{
\fill[red!75!black] (\x,{0.22*\x^4-1.1*\x^2+2}) circle (2.2pt);
\draw[dashed,black!35] (\x,0) -- (\x,{0.22*\x^4-1.1*\x^2+2});
}
\node[align=center] at (-1.75,2.75) {concave up\\$f''>0$};
\node[align=center] at (0,1.25) {concave down\\$f''<0$};
\node[align=center] at (1.75,2.75) {concave up\\$f''>0$};
\node[below] at (-0.913,0) {$x_{1}$};
\node[below] at (0.913,0) {$x_{2}$};
\end{tikzpicture}
\end{document}
```
%%
The labels on a graph must be read by behavior, not merely by whether a point lies on an axis:
![[tikz-7e9482be848c.svg]]
%% tikz-source
```latex
\begin{document}
\begin{tikzpicture}[x=1.25cm,y=0.8cm,line cap=round]
\draw[->,black!45] (-0.2,0) -- (6.1,0) node[right] {$x$};
\draw[->,black!45] (0,-1.5) -- (0,3.6) node[above] {$y$};
\draw[very thick,blue] plot[smooth] coordinates {(0.35,1.2) (0.75,2.25) (1.0,2.65) (1.45,2.15) (2.05,1.55) (2.65,0) (3.25,-1.05) (3.8,0.1) (4.55,2.35) (5.25,2.95) (5.75,2.45)};
\foreach \x/\y/\lab in {1/2.65/x_1,1.45/2.15/x_2,2.05/1.55/x_3,2.65/0/x_4,3.25/-1.05/x_5,5.25/2.95/x_6}{
\fill[red!75!black] (\x,\y) circle (2pt);
\draw[dashed,black!30] (\x,0) -- (\x,\y);
\node[below] at (\x,0) {$\lab$};
}
\end{tikzpicture}
\end{document}
```
%%
Here $x_{1}$ and $x_{6}$ are local maximum points, $x_{5}$ is a local minimum point, $x_{2}$ and $x_{3}$ are inflection points, and $x_{4}$ is only a zero of $f$.
> [!Practice]
> Marsden & Weinstein, pp. 162–163, selected exercises from $\S 3.3$.