MATHEMATICAL ANALYSIS 06
Extreme Value Theorem
Continuity on a closed bounded interval guarantees that maximum and minimum values are actually attained.
The theorem
If \(f\) is continuous on \([a,b]\), then there exist points \(c,d\in[a,b]\) such that
\[f(c)\le f(x)\le f(d)\qquad\text{for every }x\in[a,b].\]
Thus \(f(c)\) is an absolute minimum and \(f(d)\) an absolute maximum.
Why the hypotheses matter
On an open interval, an extreme value may be approached without being reached. A discontinuity can also destroy attainment.
Connection to optimisation
For a differentiable function, interior extrema are sought among critical points where \(f'(x)=0\), but endpoints must also be checked. The EVT explains why a global maximum and minimum exist under its hypotheses.