Problem Involving Maximizing the Ratio of Integrals

  • Thread starter Thread starter hartran
  • Start date Start date
  • Tags Tags
    Integrals Ratio
Click For Summary
The discussion focuses on maximizing the ratio of integrals involving functions a(x), b(x), and c(x). It is established that setting b(x) as a delta function at the maximum point of a(x)/c(x) achieves optimal results. However, the challenge arises when b(x) must remain greater than zero and cannot be a delta function, with a maximum value of C. A proposed solution is to define b(x) as C within a specific interval around the maximum point and zero outside of it, with the interval width being 1/C. This approach aims to approximate the delta function while adhering to the constraints on b(x).
hartran
Messages
1
Reaction score
0
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?
 
Mathematics news on Phys.org
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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Undergrad Error propagation of a variable for an integral
  • · Replies 6 ·
Replies
6
Views
5K
High School Mass estimate only through mass rate of change
  • · Replies 4 ·
Replies
4
Views
2K
MHB Calculating Integral with Simpson's Rule for Error < $0.5\cdot 10^{-3}$
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Gradient descent algorithm and optimizers
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Calculus of Variations on Kullback-Liebler Divergence
  • · Replies 3 ·
Replies
3
Views
2K
High School Calculating the perfect tennis shot using vectors
  • · Replies 2 ·
Replies
2
Views
2K
MHB Determining c in Quadratic Function Turning Point
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Summing over continuum and uncountable numerocities
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Integration with different infinitesimal intervals
  • · Replies 31 ·
2
Replies
31
Views
4K