#math/calculus Let $f$ be defined on an interval $I$. A number $M$ is the **maximum value** of $f$ on $I$ if $f(x)\le M$ for every $x\in I$ and $f(x_{0})=M$ for at least one $x_{0}\in I$. The point $x_{0}$ is a **maximum point** (or maximizer). Likewise, $m$ is the **minimum value** if $f(x)\ge m$ throughout $I$ and $f(x_{1})=m$ for some $x_{1}\in I$. The value and the point are different objects: a function can attain one maximum value at several maximum points. In the temperature graph below, the maximum value is $26^\circ$ while the maximum points are $t=14$ and $t=15.5$. ![[tikz-79bcfde2bc61.svg]] %% tikz-source ```latex \begin{document} \begin{tikzpicture}[x=0.48cm,y=0.35cm,line cap=round] \draw[->,black!45] (0,15) -- (19,15) node[right] {$t$}; \draw[->,black!45] (0,15) -- (0,28) node[above] {$T$}; \draw[very thick,blue] plot[smooth] coordinates {(0,16) (4,18) (8,20) (11,22.5) (12,24) (13,23.3) (14,26) (14.8,25.1) (15.5,26) (16.3,24.8) (18,23.3)}; \foreach \x in {14,15.5}{ \draw[dashed,black!40] (\x,15) -- (\x,26); \fill[red!75!black] (\x,26) circle (2.2pt); } \draw[dashed,black!30] (0,26) -- (15.5,26); \node[left] at (0,26) {$26$}; \node[below] at (14,15) {$14$}; \node[below] at (15.5,15) {$15.5$}; \end{tikzpicture} \end{document} ``` %% An upper or lower bound need not be attained. When it is not attained, it is not a maximum or minimum value; see [[3B2 - Min-Max on Chosen Intervals]].