Im not sure about when to apply b. Would the actual function include b or would it be multiplied by b after integration has been carried out?
If I can leave b out then it is simply a case of showing f(x)<xf(x) which is not so difficult.
This problem is really confusing me.
Ill give this another try then:
Case 1: Both integrals diverge thus both limits are infinifty and thus they are equal
b\int_b^∞ f(x) dx = \int_b^∞ xf(x) dx
Case 2: Both integrals converge.
b\int_b^∞ f(x) dx ≤ \int_b^∞ xf(x) dx
Would it work to move the b over to the...
Just to make sure you saw in the OP: f(x) >eq 0
Ok so:
- Both diverge to infinity means they are equal
- b-function converges x-function diverges means x-function is always bigger since of course it diverges
- both functions converge means x is greater than the factor b thus the...
Ok I have gone back and rethought the problem. If the function has an integral this must mean it converges and thus converges to 0. In this case the function will be integrated either with or without the additional x in both cases resulting in a continuous convergent to 0 function. However the...
Yes I see what you are saying. Think the last proof could work because we are assuming the function works for an improper integral. This would mean that when taking b as a high value for example would only make the original value smaller (the integrated function needs to go to a real number for...
Well with it I mean the function f(x). If it converges then it will have a maximum turning point otherwise if it diverges it should have a minimum turning point.
bf(x) \leqxf(x) is true yet I simply don't know if that is enough. I feel I can vaguely explain the solution yet I have no way of...
Homework Statement
Prove or disprove:
b\int_b^∞ f(x) dx ≤ \int_b^∞ xf(x) dx
for any b≥0 and f(x)≥0Homework Equations
N/A
The Attempt at a Solution
Ok this question has caused me quite some problems. I have come to the conclusion that this needs to be proven rather than disproven...