One common approach for "maxmin" problems, i.e., of the form
maximimize (over $\{x_i\}$) $\min \{f_i(x_i)\}$,
is to introduce an additional variable, say $t$, and reformulate the problem as
maximize (over $t$ and $\{x_i\}$) $t$, subject to $f_i(x_i)\geq t, \forall i$.
Example:
f[x_, y_, z_] := x + 2 y - 5 z;
g[x_, y_, z_] := 4 x - y + 9 z;
FindMaximum[{t,
f[x, y, z] >= t && g[x, y, z] >= t && 0 <= x <= 0.1 &&
0 <= y <= 10 && 0 <= z <= 1}, {{t, 0}, {x, 0}, {y, 0}, {z, 0}}]
{4.63333, {t -> 4.63333, x -> 0.1, y -> 4.76667, z -> 1.}}
P.S.: I imposed additional (arbitrary) constraints on the range of values of $x,y,z$ to make the objective function bounded.
FindMaxValue
andFindMinValue
in the documentation. $\endgroup$