Question

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

Answer 1

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)`.