The function minimum problem and applies to be first simple the introduction next circular function the extreme value existence condition, next mainly in the function of many variables extreme value's related theory's foundation, using the quadratic form positive definiteness and the negative definiteness discussion function of many variables's extreme value distinguished that the law as well as asks the function of many variables with the lagrange multiplicator law the condition extreme value, finally expounds one Yuan separately through the model illustration topic, function of many variables extreme value application in reality.