Question Number 571 by 123456 last updated on 30/Jan/15
![proof or given a counter example: if f,g are continuos into [a,b] and g never change sign into [a,b] then ∃c∈[a,b] such that ∫_a ^b f(x)g(x)dx=f(c)∫_a ^b g(x)dx](Q571.png)
$${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$$\:{if}\:{f},{g}\:{are}\:{continuos}\:{into}\:\left[{a},{b}\right]\:{and} \\ $$$${g}\:{never}\:{change}\:{sign}\:{into}\:\left[{a},{b}\right]\:{then} \\ $$$$\exists{c}\in\left[{a},{b}\right]\:{such}\:{that} \\ $$$$\underset{{a}} {\overset{{b}} {\int}}{f}\left({x}\right){g}\left({x}\right){dx}={f}\left({c}\right)\underset{{a}} {\overset{{b}} {\int}}{g}\left({x}\right){dx} \\ $$
Answered by prakash jain last updated on 30/Jan/15
![Since g does not change sign ∫_a ^b g(x)≠0 Hence we can write k=((∫_a ^b f(x)g(x)dx)/(∫_a ^b g(x)dx)) ...(i) Let say m=min f(x) x∈[a,b] Let say M=max f(x) x∈[a,b] case I: g(x)>0 k>M ⇒f(x)g(x)<Mg(x) then RHS in (i) less than k. k<m ⇒f(x)g(x)>mg(x) then RHS in (i) more than k. case II: g(x)<0 k>M ⇒f(x)g(x)>Mg(x) then RHS in (i) more than k. k<m ⇒f(x)g(x)<mg(x) then RHS in (i) less than k. Since LHS =k k∈f(x) x∈[a,b] or k=f(c) for some c∈[a,b]](Q575.png)
$$\mathrm{Since}\:{g}\:\mathrm{does}\:\mathrm{not}\:\mathrm{change}\:\mathrm{sign}\:\int_{{a}} ^{{b}} {g}\left({x}\right)\neq\mathrm{0} \\ $$$$\mathrm{Hence}\:\mathrm{we}\:\mathrm{can}\:\mathrm{write} \\ $$$${k}=\frac{\underset{{a}} {\overset{{b}} {\int}}{f}\left({x}\right){g}\left({x}\right){dx}}{\int_{{a}} ^{{b}} {g}\left({x}\right){dx}}\:\:\:\:\:\:\:...\left(\mathrm{i}\right) \\ $$$$\mathrm{Let}\:\mathrm{say}\:{m}=\mathrm{min}\:{f}\left({x}\right)\:{x}\in\left[{a},{b}\right] \\ $$$$\mathrm{Let}\:\mathrm{say}\:{M}=\mathrm{max}\:{f}\left({x}\right)\:{x}\in\left[{a},{b}\right] \\ $$$$\mathrm{case}\:\mathrm{I}:\:{g}\left({x}\right)>\mathrm{0} \\ $$$${k}>{M}\:\Rightarrow{f}\left({x}\right){g}\left({x}\right)<{Mg}\left({x}\right)\:\mathrm{then}\:\mathrm{RHS}\:\mathrm{in}\:\left(\mathrm{i}\right)\:\mathrm{less}\:\mathrm{than}\:{k}. \\ $$$${k}<{m}\:\Rightarrow{f}\left({x}\right){g}\left({x}\right)>{mg}\left({x}\right)\:\mathrm{then}\:\mathrm{RHS}\:\mathrm{in}\:\left(\mathrm{i}\right)\:\mathrm{more}\:\mathrm{than}\:{k}. \\ $$$$\mathrm{case}\:\mathrm{II}:\:{g}\left({x}\right)<\mathrm{0} \\ $$$${k}>{M}\:\Rightarrow{f}\left({x}\right){g}\left({x}\right)>{Mg}\left({x}\right)\:\mathrm{then}\:\mathrm{RHS}\:\mathrm{in}\:\left(\mathrm{i}\right)\:\mathrm{more}\:\mathrm{than}\:{k}. \\ $$$${k}<{m}\:\Rightarrow{f}\left({x}\right){g}\left({x}\right)<{mg}\left({x}\right)\:\mathrm{then}\:\mathrm{RHS}\:\mathrm{in}\:\left(\mathrm{i}\right)\:\mathrm{less}\:\mathrm{than}\:{k}. \\ $$$$\mathrm{Since}\:\mathrm{LHS}\:={k} \\ $$$${k}\in{f}\left({x}\right)\:{x}\in\left[{a},{b}\right] \\ $$$$\mathrm{or}\:{k}={f}\left({c}\right)\:\mathrm{for}\:\mathrm{some}\:{c}\in\left[{a},{b}\right] \\ $$