f = Exp[x/3]*(x^2+3*y^2-2*y*z-z^2+z-y+5)

gradijent = D[f,{{x,y, z}}]

Tacke = Solve[gradijent == 0,{x,y, z}]

Hodf= Simplify[D[f,{{x,y,z},2}]]

HodfM = MatrixForm[Hodf]

HodfuTacke = Hodf  /. Tacke

minf = Map[PositiveDefiniteMatrixQ,HodfuTacke,1]

maxf = Map[PositiveDefiniteMatrixQ,-HodfuTacke,1]