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Through Sin[x]^6 + Cos[x]^6 // TrigReduce operation, we can know that the minimum positive period of Sin[x]^6 + Cos[x]^6 is Pi/2

But

Periodic`PeriodicFunctionPeriod[Sin[x]^6 + Cos[x]^6, x]

returns Pi.Refer to this link.

How can MMA solve the minimum positive period of a function by a common method?

J. M.'s missing motivation
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    from help on FunctionPeriod it says : gives a period and not minimum period. So $\pi$ is valid period. Also comment there by JM in the link you posted says Another possible weakness of the function is that it does not always return the minimal period so may be one should call TrigReduce first and then call FunctionPeriod on the result. Then it does return Pi/2 in this case. May be WRI should fix this function so it returns minimum period. – Nasser Jan 20 '20 at 07:15

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