Help maximizing this functional please (it's simple, i think)

[enfield]

the functional is $$\int_0^b{\frac{f(b)-f(x)}{b-x}},$$. Another thing is it has defined endpoints. $$f(0)=0$$ and $$f(b) \in (0,10)$$ and also $$f'(x)>0.$$ oh and $$b \in (0,\infty)$$

The euler-lagrange equation didn't give me anything helpful (not sure why..). One thing that is clear when looking at the functional is that very concave functions seem to maximize it, and very convex functions seem to minimize it.

so maybe the maxima/minima are (in the limit) the two halves of the rectangle defined by $$(0,0), (b,0), (b, f(b)), (0, f(b))$$. maybe.

 

[jvicens]

I would like to provide some insights on the functional and its properties. First, the functional is known as the average value of a function over a certain interval. In this case, the interval is from 0 to b. This means that the functional is a measure of the overall behavior of the function f(x) over this interval.

The defined endpoints of f(0)=0 and f(b)∈(0,10) tell us that the function must start at 0 and end somewhere between 0 and 10. This gives us some constraints on the possible behavior of the function within the interval.

The condition f'(x)>0 tells us that the function is increasing over the entire interval. This means that the function is always getting larger as x increases. This is consistent with the observation that very concave functions (ones that curve downwards) seem to maximize the functional, while very convex functions (ones that curve upwards) seem to minimize it.

The fact that the Euler-Lagrange equation did not provide any helpful information is not surprising. The Euler-Lagrange equation is used to find the critical points of a functional, but it does not guarantee that these critical points are maxima or minima. In this case, it seems that the critical points may not be maxima or minima, but rather the two halves of the rectangle defined by (0,0), (b,0), (b, f(b)), (0, f(b)).

Overall, the functional and its properties can give us some insights into the behavior of the function f(x) over the interval (0,b). However, further analysis and investigation may be needed to fully understand the behavior and potential maxima or minima of the functional.