MHB Relative Extrema vs Absolute Extrema

[harpazo]

In basic terms, what is the main difference between relative extrema and absolute extrema? I know that absolute extrema is more involved but why is this the case?

 

[mathmari]

A relative maximum (minimum) is the greatest (smallest) in its neighborhood but an absolute maximum (minimum) is the greatest (smallest) anywhere (in the domain). For example:

View attachment 6470

In the picture we see that the function has $3$ relative minima and an absolute minimum, which is smallest of all relative minima. It also has $2$ relative maxima, but it has no absolute maximum, since at the boundaries the function goes to $+\infty$.
In general, to find an absolute extrema, besides the critical points we have to check also the poles of the function and the boundaries of its domain.

 

[harpazo]

A relative maximum (minimum) is the greatest (smallest) in its neighborhood but an absolute maximum (minimum) is the greatest (smallest) anywhere (in the domain). For example:



In the picture we see that the function has $3$ relative minima and an absolute minimum, which is smallest of all relative minima. It also has $2$ relative maxima, but it has no absolute maximum, since at the boundaries the function goes to $+\infty$.
In general, to find an absolute extrema, besides the critical points we have to check also the poles of the function and the boundaries of its domain.

Very cool. Nice picture.