Discuss the below:
Let f(x), g(x) be functions defined on a closed bounded interval [a, b] such that the following conditions hold: g is differentiable on [a, b].
There are positive constants a, b such that g(x) = a*f(x) - b*(dg/dx).
f(x) > 0 for all x in [a, b]
g(x) >= 0 for all x in [a, b]
g(a) > 0
Q1: At how many points of [a, b] could g(x) possibly be equal to 0:
i. none
ii. just one
iii. finitely many but more than one
iv. countably infinitely many
v. uncountably many
Q2: If the answer to Question 1 is something other than "i," at what point(s) of [a, b] could g(x) possibly be equal to 0?