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We systematically investigate different versions of variational perturbationtheory by forcing not only the first or second but also higher derivatives ofthe approximant with respect to the variational parameter to vanish. The choiceof the highest derivative version turns out to be the most successful one forapproximating the ground-state energy of the anharmonic oscillator. It istherefore used to determine the critical exponent alpha of the specific heat insuperfluid 4He in agreement with the value measured in recent space shuttleexperiments.Comment: The article is a contribution to the book "Fluctuating Paths and  Fields - Dedicated to Hagen Kleinert on the Occasion of His 60th Birthday",  Eds. Wolfhard Janke, Axel Pelster, Hans-Juergen Schmidt, and Michael Bachmann  (World Scientific, Singapore, 2001), p. 347-36

THE HIGHEST-DERIVATIVE VERSION OF VARIATIONAL PERTURBATION THEORY