The problem involves two functions, f and g, defined on the real numbers, with specific properties that relate them. The goal is to prove that the derivative of g, denoted g'(x), is equal to g(x) for all x. The discussion revolves around the implications of the given properties of g and the limit definition of the derivative.
Participants are actively engaging with the problem, with some providing hints and affirmations regarding the use of the limit definition. There is a recognition of the importance of the properties of g in the derivation process, and some participants express confidence in the approach being taken.
There is an emphasis on the three properties of g that must be considered while applying the limit definition of the derivative. The original poster expresses uncertainty about how to start the problem, indicating a need for guidance in understanding the implications of the properties provided.
Just use the "limit" definition of the derivative.drawar said:Homework Statement
Let f and g be two functions on ℝ such that:
1. g(x) = xf(x) + 1 for all x,
2. g(x + y) = g(x)g(y) for all x,y,
3. \mathop {\lim }\limits_{x \to 0} f(x) = 1.
Prove that: g'(x) = g(x) for all x.
Homework Equations
The Attempt at a Solution
Sorry for asking this question without showing my workings but I just don't know how to get started. Any hints would be much appreciated. TIA!
SammyS said:Just use the "limit" definition of the derivative.
drawar said:So I have to prove \mathop {\lim }\limits_{h \to 0} \frac{{g(x + h) - g(x)}}{h} = g(x)?
SammyS said:Yes.
It works out quite nicely.