#f(x) = tanx# on #[-2pi, 2pi]# does not satisfy the hypotheses of the Mean Value Theorem. It also does not satisfy the conclusion.
The hypotheses call for a function that is continuous on #[a,b]# (and differentiable on #(a,b)#).
Tangent is undefined for odd multiples of #pi/2#, so #tanx# is not continuous on #[-2pi,2pi]#.
#f(x) = tanx# on #[-2pi, 2pi]# does not satisfy the conclusion of the Mean Value Theorem.
#f'(x) = sec^2x# and #(f(2pi)-f(-2pi))/(2pi-(-2pi)) = (0-0)/(4pi) = 0#.
Since #sec^2 x >= 1# for all #x#, there is no #c# anywhere for which #f'(c) = (f(2pi)-f(-2pi))/(2pi-(-2pi)) #
