why doesn't "coeff" command work on this equation?

f=zin-z2(z3(z1(zin+I*z1*tan(beta1*d1))+I*z3*(tan(beta3*d3))(z1+I*zin*tan(beta1*d1)))
+I*z2*(tan(beta2*d2))(z3*(z1+I*zin*tan(beta1*d1))+I*z1(zin+I*z1*tan(beta1*d1))*tan(beta3*d3)))/(z2(z3(z1+I*zin*tan(beta1*d1))
+I*z1(zin+I*z1*tan(beta1*d1))*tan(beta3*d3))+I*z3(z1(zin+I*z1*tan(beta1*d1))+I*z3*(tan(beta3*d3))(z1+I*zin*tan(beta1*d1)))*tan(beta2*d2))

a0 := coeff(f, zin, 2);
b0 := coeff(f, zin, 1);
c0 := coeff(f, zin, 0);

how is it possible to identify the independent variable that maximize(or minimize) a function?
is it important to know the kind of the function?

Is it possible to specify the independent variable that maximize a function just via it's plot?

hi friends!
I have a 2*2 matrix(namely Mcell), whose elements vary as a complex function of "nu".
for each cycle of a "for" loop, h[j] := abs(Re(Mcell[2, 2])), whose diagram is periodic and has many crosses with y=1.
i just wanted the first two points(their difference) and used

L := Roots(h[j]-1, nu = 1 .. something, numeric); and
w[j] := L[2]-L[1];

in a "for" loop.

i check the answer, they are all true(w gets 20 values in 20 iterations).

how can i plot 20 pairs of (x,y) data,displaying both dots and one of available commands of CurveFitting Package in one diagram? for example if we have L:=[1,2,...,20]: and w:=[23,24,25,...,42]: plot dots:[1,23],[2,24],...,[20,42](but without writing these 20 pairs manual!) together with a fitting line in one diagram.

how can i get the inetersection point (a real number of x axis) of a complex plot and a real plot? and how to assign them respectively to an array?

for example the junctions of lines:

d1:=(sec(I*(x^2-2)))(exp(I*x)*sin(2*x)+cos(I*(2*x-3)));       and

d2:=0.5;  by  plot([d1, d2], x = -5 .. 5);