28. (a) The derivative of P (E) is

-1 d -1 1
F F
= .
2 dE e(E-E )/kT 2 kT e(E-E )/kT
F F
e(E-E )/kT +1 e(E-E )/kT +1
Evaluating this at E = EF we readily obtain the desired result.
(b) The equation of a line may be written y = m(x - xo) where m is the slope (here: equal to -1/kT ,
from part (a)) and xo is the x-intercept (which is what we are asked to solve for). It is clear that
P (EF ) = 2, so our equation of the line, evaluated at x = EF , becomes 2 = (-1/kT )(EF - xo),
which leads to xo = EF +2kT .