TITLE: Find the value of $k$
QUESTION [0 upvotes]: If $f:[1,2]\mapsto \Bbb R$ is a non-negative integrable function such that $\int_1^2\dfrac{f(x)}{\sqrt x}\operatorname{dx}=k\int _1^2f(x)\operatorname{dx}\neq 0$,then find $k$.
I was thinking to use integration by parts which is not working.Any hints will be helpful.
Please provide some hints only.

REPLY [0 votes]: by the mean value theorem for the integrals, you have $$0=\int_1^2 f(x)\left(\frac1{\sqrt x} - k\right)\, dx = \left(\frac 1{\sqrt c} -k\right)\int_1^2f(x)\, dx \text{ for some $c$ between $1$ and $2.$}$$
without any further information on $f,$ all we can conclude is that $$\frac 1{\sqrt 2 } < k < 1. $$
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