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Hi, Can someone help me solve this question. I wrote IIT JEE2001 and passed. But I am unable to solve this question now :(. I was trying out questions for nostalgic reasons. Let f(x), x > 0, be a non-negative continuous function, and let F(x) = ∫ 0 x f(t) dt, x > 0. If for some c > 0, f(x) cF(x) for all x > 0, then show that f(x) = 0 for all x > 0.

Hi,


  Can someone help me solve this question. I wrote IIT JEE2001 and passed. But I am unable to solve this question now :(. I was trying out questions for nostalgic reasons.


 


 


Let f(x), x > 0, be a non-negative continuous function, and


let F(x) = ∫0x f(t) dt, x > 0. If for some c > 0, f(x) < cF(x) for all x > 0,


then show that f(x) = 0 for all x > 0.

Grade:

1 Answers

AJIT AskiitiansExpert-IITD
68 Points
13 years ago

Dear Ram ,

visualise it in terms of graphs. f(x) is nonnegative continuous function means graph will be on or above the x-axis. now , F(x) is just area under the curve upto the point x.  so , F(x) >= f(x) as area is always greater than the funciton itself.  i think you have mistyped the relation f(x) < cF(x) ,   f(x) > cF(x) as then actually f(x) = F(x)  = 0   for any c

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