Finding Continuous Functions to Solve a Riemann-Stieltjes Problem

[cinlef]

Am studying for an analysis final, and I've come across a problem that's left me totally stumped... I feel like I must be missing something. Could anyone give me a hand here?

Homework Statement

Assume g(x) is strictly increasing and continuous on [a,b]. Suppose g(b)-g(a)=1. Find all continuous functions, f such that, let
int f dg=1
int f² dg=1
Where the integrals are taken from a to b (apologies for the lack of Latex skill)

The Attempt at a Solution

Honestly not sure where to start. Any tips?

 

[mighty2000]

Dear fellow student,

I can understand how this problem may seem challenging at first glance. However, there are a few key concepts that can help guide your approach.

First, remember that the integral of a function f over an interval [a,b] can be interpreted as the area under the curve of f over that interval. This means that the integral of f dg represents the area between the graph of f and the x-axis over the interval [a,b].

Next, recall that the integral of a function g over an interval [a,b] can be interpreted as the net change of g over that interval. In other words, if g is a continuous and strictly increasing function, then the integral of g dg represents the change in the y-values of g over the interval [a,b].

Now, let's apply these concepts to the given problem. We know that g(b)-g(a)=1, which means that the net change of g from a to b is 1. This also means that the area between the graph of g and the x-axis over the interval [a,b] is also 1.

From the given information, we can also see that the integral of f dg is equal to 1. This means that the area between the graph of f and the x-axis over the interval [a,b] is also 1. In other words, the net change of f from a to b is also 1.

Now, think about what this means for the graph of f. Since the net change of f from a to b is 1, this means that f must be increasing over the interval [a,b]. Additionally, since the area between the graph of f and the x-axis is also 1, this means that f must be above the x-axis over the entire interval [a,b]. This also implies that f must be strictly positive over the interval [a,b].

Using these observations, you can start to think about what types of functions f could be. For example, f could be a linear function with a positive slope, or it could be a polynomial function with positive coefficients. Can you think of any other types of functions that would satisfy the given conditions?

I hope this helps to guide your thinking. Remember to use the concepts of net change and area under the curve to guide your approach. Good luck with your final!

(Fellow Scientist)