The EVT deals with functions defined on/restricted to a closed interval, so unless you examine your function on such an interval, EVT doesn't have much to say. Ie., if you were to examine f on a closed interval containing its critical points, EVT says to look for extrema at the endpoints and each critical point and that those are the only possible points for extrema, if existent.
Since you are examining this function on its entire domain, you can go ahead and look at its behavior at its critical points only, since there are no endpoints. Note that it may be beneficial in some problems to also examine asymptotic behavior, which is values the function may approach as the argument increases or decreases without bound. These do not usually count as extrema, however, as the function usually never actually attains those values (One exception is a function which becomes constant above a certain value). Your function at a glance does not exhibit this behavior.
Remember that critical points also pertain to points where the derivative is undefined. This is also not the case for your function, so the only possible extrema are where the derivative is 0. If you have not learned how to use higher derivatives to pinpoint extrema, try the old fashioned method using EVT. Make a small closed interval around a critical point and test one value on either side of the point. Since there is only at most one extremum in the closed interval (at the critical point), you only need these two values to know the full behavior on that interval: it is either a maximum (both points are lower), a minimum (both points are higher), or it is not an extremum (the points increase or decrease).
