#math/calculus To sketch a graph from a formula, organize the information before drawing: 1. Find the domain, symmetry, intercepts, asymptotes, and end behavior. 2. Solve $f'(x)=0$ and note where $f'$ is undefined. Use the sign of $f'$ to mark increasing/decreasing intervals and local extrema. 3. Solve $f''(x)=0$ and note where $f''$ is undefined. Use its sign to mark concavity and inflection points. 4. Plot the key points and connect them in a way consistent with every interval test. For $f(x)=x^{3}-3x$, we have $f'(x)=3(x^{2}-1)$ and $f''(x)=6x$. Thus $f$ has a local maximum at $(-1,2)$, a local minimum at $(1,-2)$, and an inflection point at $(0,0)$. | Interval | Sign of $f'$ | Behavior | Sign of $f''$ | Concavity | |---|---:|---|---:|---| | $(-\infty,-1)$ | $+$ | increasing | $-$ | down | | $(-1,0)$ | $-$ | decreasing | $-$ | down | | $(0,1)$ | $-$ | decreasing | $+$ | up | | $(1,\infty)$ | $+$ | increasing | $+$ | up | ![[tikz-b47e3795a024.svg]] %% tikz-source ```latex \begin{document} \begin{tikzpicture}[x=1.35cm,y=0.62cm,line cap=round] \draw[->,black!45] (-2.5,0) -- (2.55,0) node[right] {$x$}; \draw[->,black!45] (0,-4.8) -- (0,4.8) node[above] {$y$}; \draw[very thick,blue,domain=-2.15:2.15,samples=120] plot (\x,{\x^3-3*\x}); \fill[red!75!black] (-1,2) circle (2.3pt) node[above left] {$(-1,2)$}; \fill[red!75!black] (1,-2) circle (2.3pt) node[below right] {$(1,-2)$}; \fill (0,0) circle (2pt) node[above right] {$(0,0)$}; \end{tikzpicture} \end{document} ``` %% > [!Practice] > Marsden & Weinstein, p. 175, selected exercises from $\S 3.4$.