Extreme Value Theorem –
If \(f\) is continuous on a closed interval \([a,b],\)
there must exist an absolute extremum of \(f\) on that interval.
The extrema of \(f\) can be found
among all the values of \(f\) on the boundary of its domain
along with all the value of \(f\)
at the critical points on the interior of its domain.
The problem of determining and computing these extrema
is called an optimization problem.
An optimization problem statement has this standard template:
\[ \text{``Minimize/Maximize [}\mathit{function}\text{] subject to [}\mathit{constraints}\text{].''} \]
To solve an optimization problem in this standard form:
- Write down a formula for the \(\mathit{function};\) it might contain more than one independent variable. Then write down formulas for any \(\mathit{constraints},\) and use those constraints to re-write the \(\mathit{function}\) as a single-variable function \(f.\)
- Determine any points on the boundary of the domain of \(f\) (if it has a boundary), and any critical points of \(f.\)
- Evaluate \(f\) at each of those critical points \(c,\) and at any points on the boundary of its domain (e.g. \(f(a)\) and \(f(b)\)). Among all of those output values, the largest will be the absolute maximum value of \(f\) and the smallest will be the absolute minimum value of \(f.\)