#math/calculus For $f(x)=x^{2}+1$, the interval determines whether a bound is attained. A `maximum point` or `minimum point` is an input $x$; the corresponding `maximum value` or `minimum value` is $f(x)$. | Interval $I$ | Maximizer(s) | Maximum value | Minimizer(s) | Minimum value | |---|---:|---:|---:|---:| | $(-\infty,\infty)$ | none | none | $0$ | $1$ | | $(0,\infty)$ | none | none | none | none | | $(-1,1)$ | none | none | $0$ | $1$ | | $[-1,1]$ | $-1,1$ | $2$ | $0$ | $1$ | | $(0,1]$ | $1$ | $2$ | none | none | | $[\tfrac12,1]$ | $1$ | $2$ | $\tfrac12$ | $\tfrac54$ | | $(-2,1]$ | none | none | $0$ | $1$ | | $[-2,1)$ | $-2$ | $5$ | $0$ | $1$ | Open endpoints explain many missing extrema. On $[2,4)$, $f$ has no maximum: its values approach $17$ but never equal it. On $(3,78]$, $f$ has no minimum: its values approach $10$ but never equal it. > [!Insight] > A supremum or infimum is a best bound. It becomes a maximum or minimum only when the function actually attains it.