How to prove a given statistical test has the greatest power It is very conventional in evaluating the null hypothesis to consider the distribution of the mean of a sample of some size if the null hypothesis were true and to compare the mean of your own sample to that distribution.  (At least if the underlying distribution is thought to be normal) But why, even given the normality assumption, do statisticians generally compare the distribution of the mean to the sample mean as opposed to comparing the distribution of some other statistic, say the median or the minimum, to the median (or minimum) of the sample.
I believe the answer to my question is that the mean is a powerful statistic.  If the sample indeed did not come from the null hypothesis distribution, then the mean test is likely to pick it up.  It might take a much greater effect for the median or min test to obtain an adequately low p-value?  But, how do we KNOW that there are not better statistics than the mean?  Is there a proof out there in the literature?  Or how would one possibly go about proving it?

You seem to be interlacing two questions: 1) Does this sample come from a distribution with mean $\mu$? and 2) Does this sample come from a specific distribution (e.g. a normal distribution with mean $\mu$ and standard deviation $\sigma$). If you want to test whether your sample comes from a particular distribution, or test whether two samples come from the same (unknown) distribution, there are more powerful tests than comparison of means, like Kolmogorov-Smirnov test that you can look up online (that can reject the null that a sample comes from a distribution even if the means are equal). Such a test basically compare all quantiles of the two distributions, not just the mean or median or min or max.
