We give a new proof of the theorems on the maximum entropy principle in
Tsallis statistics. That is, we show that the q-canonical distribution
attains the maximum value of the Tsallis entropy, subject to the constraint on
the q-expectation value and the q-Gaussian distribution attains the maximum
value of the Tsallis entropy, subject to the constraint on the q-variance, as
applications of the nonnegativity of the Tsallis relative entropy, without
using the Lagrange multipliers method. In addition, we define a q-Fisher
information and then prove a q-Cram\'er-Rao inequality that the q-Gaussian
distribution with special q-variances attains the minimum value of the
q-Fisher information