Maximizing integral ratio with bounded b(x)

[hartran]

The ratio of integrals

∫〖a(x) b(x)dx〗/ ∫c(x) b(x)dx

can be maximized by choosing b(x) equal to the delta function at the point where a(x)/c(x) is a maximum.

Can anyone provide the solution for choosing b(x) when b(x) cannot equal the delta function, b(x) is greater than zero with a maximum value of C, and a(x) and c(x) are both positive over the integration interval and also monotonically increasing?

 

[mathman]

I suspect the best you could do is take an interval around the x where a(x)/c(x) is maximum. The width of the interval would be 1/C, while b(x) = C in the interval, and = 0 otherwise.