#F(z) = z^2-z-2#
#F(z)# is a quadratic function with real coefficients.
I'm not claiming that the following constitutes an "essay". Nor would I know how to write one on #F(z)#. Also, I do not understand what is meant by, "Need to relate what?". Nevertheless, I will set out some of the attributes of #F(z)# below.
#F(z)# factorizes into #(z-2)(z+1)#
#:.# the zeros of #F(z)# are #z=2 or -1#
The graph of #F(z)# will be a parabola with axis of symmetry #z=1/2#
Since the coefficient of #z^2# is #>0 -> F(z)# will have a minimum values on its axis of symmetry.
Therefore, the minimum value of #F(z)# is #F(1/2)#
Thus, #F_min = F(1/2) = 1/4-1/2-2 = -9/4#
The graph of #F(z)# is shown below - where #z# is the horizontal axis.
graph{x^2-x-2 [-3.798, 4.97, -2.53, 1.855]}
As an aside, using the nomenclature #F(z)# usually indicates that #z# is a complex variable. In this case, however, #{F(z),z} in RR#
