Global maximum/ minimum of a function of more than one variable.

Question

Please, can someone give me more information on how to check if points are local or global maximum/minimum. I am aware of the second derivative test of determining the local minimum/maximum. But how do I check for global min/max? An example to illustrate this would be very much appreciated. Thanks.

Answer 1

A very useful method is Sturm's principle which says that a continuous function on a compact domain (e. g. a closed interval) which is bounded below/above an has at most one local minimum/maximum must have a global minimun/maximum at this very place.

If your function in question, however, has more than one local minimum/maximum, you have to compare the values of the function at all these minima/maxima and those with minimal/maximal value among these candidates are your global minima/maxima.

In each case, you should check first, that your function is bounded below/above!

Thus, for $f:[a,b] \rightarrow \Bbb R$ differentiable, you should proceed as follows:

1. Calculate the zeros $x_1,x_2,x_3, ...$ of $f'(x)$
2. Calculate $f(x_1),f(x_2),f(x_3), ...$ as well as f(a)
3. The minimum of the values calculated in step 2 is your global minimum.