4.2 | Optimization

The Extreme Value Theorem:
If \(f\) is continuous on the closed interval \(a \leq x \leq b,\) then \(f\) has a global maximum and a global minimum on that interval.
Global Maxima and Minima on a Closed Interval: Test the Candidates
For a continuous function \(f\) on a closed interval \(a \leq x \leq b:\)

  • Find the critical points of \(f\) in the interval.
  • Evaluate the function at the critical points and at the endpoints, \(a\) and \(b.\) The largest value of the function is the global maximum; the smallest value of is the global minimum.
Global Maxima and Minima on a Open Interval or All Real Numbers
For a continuous function \(f,\) find the value of \(f\) at all the critical points and sketch a graph. Look at values of \(f\) when \(x\) approaches the endpoints of the interval, or approaches \(\pm \infty,\) as appropriate. If there is only one critical point, look at the sign of \(f’\) on either side of the critical point.

E 4.2 Exercises