  The functions of the Takagi exponential class are similar in construction to
the continuous, nowhere differentiable Takagi function described in 1901. They
have one real parameter $v\in (-1;1)$ and at points $x\in{\mathbb R}$ are
defined by the series $T_v(x) = \sum_{n=0}^\infty v^n T_0(2^nx)$, where
$T_0(x)$ is the distance between $x$ and the nearest integer point. If $v=1/2$
then $T_v$ coincides with Takagi's function. In this paper, for different
values of the parameter $v$, we study the global extremes of the functions
$T_v$, as well as the sets of extreme points. All functions of $T_v$ have a
period of $1$, so they are investigated only on the segment $[0;1]$. This study
is based on the properties of consistent and anti-consistent polynomials and
series, which the first half of the work is devoted to.
