#math/calculus
**Second Derivative Test.** Suppose $f'(x_{0})=0$ and $f''$ exists near $x_{0}$.
| Value of $f''(x_{0})$ | Conclusion |
|---|---|
| $f''(x_{0})>0$ | $x_{0}$ is a strict local minimum point |
| $f''(x_{0})<0$ | $x_{0}$ is a strict local maximum point |
| $f''(x_{0})=0$ | the test is inconclusive |
The last case gives no answer: $x^{4}$ has a minimum at $0$, $-x^{4}$ has a maximum there, and $x^{3}$ has neither, yet all three have first and second derivative equal to $0$ at the origin.
![[tikz-9878b4f9893b.svg]]
%% tikz-source
```latex
\begin{document}
\begin{tikzpicture}[x=1.05cm,y=0.75cm,line cap=round]
\begin{scope}
\draw[->,black!45] (-2.3,0) -- (2.4,0) node[right] {$x$};
\draw[->,black!45] (0,-0.2) -- (0,4.1) node[above] {$y$};
\draw[very thick,blue,domain=-1.8:1.8,samples=80] plot (\x,{\x*\x+0.55});
\draw[orange!80!black,thick] (-1.5,0.55) -- (1.5,0.55);
\fill (0,0.55) circle (2pt);
\node[below right] at (0,0.55) {$x_{0}$};
\node at (0,-0.75) {$f'(x_{0})=0,\ f''(x_{0})>0$};
\end{scope}
\begin{scope}[xshift=6.2cm]
\draw[->,black!45] (-2.3,0) -- (2.4,0) node[right] {$x$};
\draw[->,black!45] (0,-0.2) -- (0,4.1) node[above] {$y$};
\draw[very thick,red!75!black,domain=-1.8:1.8,samples=80] plot (\x,{3.4-\x*\x});
\draw[orange!80!black,thick] (-1.5,3.4) -- (1.5,3.4);
\fill (0,3.4) circle (2pt);
\node[above right] at (0,3.4) {$x_{0}$};
\node at (0,-0.75) {$f'(x_{0})=0,\ f''(x_{0})<0$};
\end{scope}
\end{tikzpicture}
\end{document}
```
%%
The same sign controls **concavity**: $f''>0$ means concave up, while $f''<0$ means concave down. If $\ell(x)=f(x_{0})+f'(x_{0})(x-x_{0})$ is the tangent-line approximation and $h=f-\ell$, then $h(x_{0})=h'(x_{0})=0$ and $h''=f''$. Thus a positive second derivative bends $f$ above its tangent near $x_{0}$; a negative one bends it below.
See also [[2A2 - Second Derivatives]] and [[3A2 - Increase and Decreasing Functions|the first derivative test]].