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I need to minimize (preferably symbolically) functional

F = Integral(f(x)*cos(x)dx)

with constraints:

1)f(x) >= 0

2) Integral(f(x)dx) = 1

3) f(0) = 1

4) f'(x) <= 0

Is there any way to solve problem like this? Could this problem be solved in principle?

Thanks in advance

Gageen0
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  • Not terribly helpful in answering your question, but this integral, is it supposed to be over positive reals? At the very least constraint 2) should be a definite integral to make sense. As for your actual question, certainly if you just use maths this problem looks very solvable, the intuition just being that cosx is a nice thing to multiply against and probably ou can do some rewriting. So it can probably be solved in principle, it looks like an elementary calculus of variations problem. I imagine something like mathematica can solve this problem, but you may need to rewrite the problem. – Countingstuff May 12 '20 at 18:10
  • Anyway I reckon the answer is 0? If you split the integral into blocks of length pi, then each of these integrals should be >= 0 as per f >=0 and f' <= 0. Then if you take f = 1 at 0, f = 1/pi on 0 to pi, f = 0 else (and make it differentiable in the usual way) you get 0. – Countingstuff May 13 '20 at 12:28

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