gingerspice81
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- Oct 31, 2005
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Let f be a continuous function with domain x>0 and let F be the function given by F(x)= x~1 f(t)dt for x>0. Suppose that F(ab)=F(a)+F(b) for all a>0 and b>0 and that F'(1)=3.
a) Find f(1)
b) Prove that aF'(ax)=F'(x) for every positive constant a
c) Use the results from parts (a) and (b) to find f(x). Justify your answer.
As a side note, "x~1" is supposed to to be the integration symbol with x as the upper limit and 1 as the lower limit (I have no clue how to do type out an integration problem).
I figured out that f(1)=3, according to the Second Fundamental Theorem of Calculus, for part (a), but I have no clue how to do parts (b) and (c). Please help!
a) Find f(1)
b) Prove that aF'(ax)=F'(x) for every positive constant a
c) Use the results from parts (a) and (b) to find f(x). Justify your answer.
As a side note, "x~1" is supposed to to be the integration symbol with x as the upper limit and 1 as the lower limit (I have no clue how to do type out an integration problem).
I figured out that f(1)=3, according to the Second Fundamental Theorem of Calculus, for part (a), but I have no clue how to do parts (b) and (c). Please help!
