Suppose $f'(x)$ exists on $[0,\infty)$, prove or disprove that: the following two integrals $\int_{0}^{+\infty}\frac{2dx}{f(x)} \ \ \text{and}\ \int_{0}^{+\infty}\frac{dx}{f(x)+f'(x)}$ have the same convergence.
The convergence of the improper integrals!
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analysis
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1What measure do you use to say that his question almost always have exclamations? :P – 2012-10-27
1 Answers
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This is not true. For a counterexample, use $f(x) = e^{-x}$ on $[0,1]$, $f(x) = e^{x}$ on $[2,\infty)$, and interpolate smoothly with a positive function on $[1,2]$. (Obviously here the problem is on $[0,1]$, not at $\infty$.)
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0I cannot understand it. We are assuming that $f'(x)\ge0$ right? How could we use $e^{-x}$? And I do not find this to be a counterexample. Could you tell me what I am missing? Thanks. – 2013-07-23