IfrelativemaximumweretakentomeanthatthevalueofthefunctionatthecriticalpointisgreaterthanorequaltothenearbyvaluesthentheconditiononthematrixSisthatitisnegativesemidefinite.
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If relative maximum were taken to mean that the value of the function at the critical point is greater than or equal to the nearby values then the condition on the matrix S is that it is negative semidefinite.