This question is probably trivial for those with decent training in math. Unfortunately, I have very limited training, and I got it over forty years ago and have forgotten a lot. I have tried getting this problem answered on other sites, but those sites seem to be designed to help students in high school or early in college math. 


My math training ended after 2 semesters of calculus, 1 semester of abstract algebra, 1 semester of probability and statistics, and 1 semester of numerical methods. Based on what I have found in wikipedia, I suspect my problem is a very simple one in the calculus of variations. Unfortunately, I know nothing about that field and had trouble understanding even the notation in the article. 


I want to determine whether the definite integral over a defined interval of ANY function that meets certain minimum constraints has a maximum value, whether that value is caclulable if it exists, and, if so, what that maximum is. If the constraints specified are insufficient, then I need to determine what additional constraints to impose. Ideally, I'd like to understand not merely what the answer is, but why that is the answer.


Given


(1) $0 < a < b$.


(2) $f(x)$ is continuous at every real.


(3) $f(x)$ can be integrated over any closed real interval.


(4) The derivative of $f(x)$ may or may not exist at every real.


(5) If $0 \leq x < y$, then $f(x) \geq f(x + y)$.


(6) $f(x) \geq g(x) = 1 /(a + b)$ if $0 \leq x \leq a$. 


(7) $0 < f(x) \leq g(x) = [(b - x)/(b^2 - a^2)]$ if $a \leq x < b$.


(8) $f(x) = g(x) = 0$ if $b \leq x$.


(9) $\int\_0^b f(x)dx = \int\_0^b g(x)dx = 1/2 $.


(10) Set $j(x) = xf(x)$.


(11) Set $k(x) = xg(x)$.


Questions:
(a) Are the constraints on $f(x)$ sufficient to set a maximum bound on $\int\_0^b j(x)dx$?


(b) If the bound exists, is it calculable?


(c) If it is calculable, what is it?


(d) If the answers to a, b, and c are negative, what are the minimum additional constraints that must be imposed on $f(x)$?


I have a hand-waving argument that says the maximum of $\int\_0^b j(x)dx$ over all possible $j$ equals $\int\_0^b k(x)dx$ (with $j$ and $k$ as above). If that is right (and I am not sure that it is), what is the argument that supports it?


I hope my question is specific enough. I apologize in advance for its probable triviality, but the high school sites do not handle this sort of question.


Jeff Morrow



Maximizing a Definite Integral Subject to Constraints
I think this isn't too bad, and your guess in below point (d) is justified. I doubt it depends on the detailed formulae. 


Suppose we concentrate on *g - f*, which is constrained to have integral 0? And by a change of origin to have a graph that is non-positive on the negative real axis, and non-negative on the positive real axis. The main issue is that multiplying it by the independent variable (*x* in the current notation) you will get a non-negative function, so a non-negative integral. I believe this is what you want.


