The essence of both anarchism and minarchism is that people function within society - voluntarily. What has to be assumed (human nature) is that humans aspire for knowledge which means that the laws of human action will be discovered and understood to a greater and greater extent over time. I mention this because society will break down if it is locked into ignorance and superstition. There are historical examples of these bleak periods.
Understanding economics is a necessary condition for progress. For example, capital is an economic reality and cannot be made into an imaginary monster by superstition if progress is to be made.
Back to social cooperation that occurs voluntarily, the anarchist is certain that everything can be done within society without any government. The minarchist cannot conceptualize circumstances which do not require some minimal amount of government for society to be stable.
Economics comes to the rescue here too! Economics is the study of the best meansto attain the ends chosen. Both the anarchists and the minarchists have similar ends in mind - peace and prosperity and justice. If the best means to achieve this is no government then the minarchist would easily accept the anarchist approach. If the best means (because it can be made to stay within specifically delineated limits) is some government then the anarchist will moderate his or her belief since the ends sought is attained in the best possible way.
The key here is the voluntary nature of choosing the best means. In the natural environment of social cooperation - which is mutually beneficial to everyone - choices are made. None of the choices are 'carved in stone' so if things don't work out as expected the choice can be reversed or changed or if new circumstances arise making other choices more likely to be most beneficial to everyone then the act of voluntarily cooperating socially will allow for amendments. If a society (the size can be decided voluntarily) decides to use as its means of administering a solution a government rather than a business and the decision is made cooperatively and voluntarily then there is no reason for anyone to object. Various means can be evaluated and any one can be chosen in a free society. This is simply a noncoercive means chosen and as long as it remains noncoercive there can be no legitimate opposition because such opposition is itself coercive under these circumstances.
Economics - the study of the best means to attain the ends - will continuously hold any and all of the means chosen accountable in the perpetual search for the best means as conditions change over time.