Some established results concerning best $L_p $-approximations are introduced, and it is shown that a minimax approximation can generally be found as the limit of a sequence of such approximations. The values of p required, however, are too large to be practicable, and the main part of this paper is devoted to the development of an extrapolation scheme which enables minimax approximations to be predicted from best $L_p $-approximations for comparatively small values of p. Numerical evidence is presented to show that the scheme works well in practice.