Graphs with prescribed extrema (on ℝ)

Text

For each item draw a different function, defined on the whole of $\mathbb{R}$ and continuous. The function must:

• (a) have both an absolute maximum and an absolute minimum;
• (b) have an absolute minimum, a finite supremum, and no relative maximum.

Solution

A construction exercise: we exhibit one valid example for each item.

(a) A function bounded on all of $\mathbb{R}$ that attains both extremes: $\boxed{f(x)=\dfrac{x}{1+x^2}}$ It is continuous on $\mathbb{R}$; it has absolute maximum $\tfrac12$ at $x=1$ and absolute minimum $-\tfrac12$ at $x=-1$; as $x\to\pm\infty$ it tends to $0$ (horizontal asymptote $y=0$). Both extremes are attained.

(b) A single valley: an absolute minimum attained at $x=0$, which then rises toward a finite value never reached. Model: $\boxed{f(x)=\dfrac{-1}{1+x^2}}$ It has absolute minimum $-1$ at $x=0$ (attained). As $x\to\pm\infty$ it tends to $0$: thus $\sup=0$ is finite but not attained, so there is no absolute maximum; and since the function is increasing for $x>0$ and decreasing for $x<0$ with no interior humps, it has no relative maximum either.