If you only have #f'(x)# is it possible to differentiate the local and global extrema of #f(x)#?

1 Answer
Oct 31, 2015

Answer:

If #f'(x)# is defined on the whole of #RR# then it is possible to reconstruct #f(x)# up to a constant offset and hence possible to distinguish local and global extrema. If not then it is not.

Explanation:

If #f'(x)# is known and defined on the whole of #RR#, then we can define

#f_1(x) = int_0^x f'(t) dt#

Then #f_1(x) = f(x) + C# for some constant #C#

Hence we can evaluate #f_1(x)# for any #x# which is a root of #f'(x) = 0# in order to distinguish local and global maxima.

On the other hand, if #f'(x)# is not defined for some values of #x in RR#, then we cannot distinguish local and global maxima.

For example, if #f'(0)# is not defined, then we do not know how the values for maxima and minima of #f(x)# for #x in (-oo, 0)# compare with maxima and minima of #f(x)# for #x in (0, oo)#.