determine-f-continue-on-a-b-wich-verify-a-b-f-x-dx-2-a-b-f-2-x-dx-

Question Number 96837 by mathmax by abdo last updated on 05/Jun/20

determine f continue on [a, b] wich verify {(\int_{a}^{b} f (x) dx)}^{2} = \int_{a}^{b} f^{2} (x) dx

Answered by mathmax by abdo last updated on 07/Jun/20

let b = t (afixed) \Rightarrow \frac{d}{dt} {(\int_{a}^{t} f (x) dx)}^{2} = \frac{d}{dx} \int_{a}^{t} f^{2} (x) dx \Rightarrow

2 \int_{a}^{t} f (x) dx f (t) = f^{2} (t) \Rightarrow 2 \int_{a}^{t} f (x) dx = f (t) \Rightarrow

{2 f}^{^{'}} (t) = f^{^{'}} (t) \Rightarrow f^{^{'}} (t) = 0 \Rightarrow f = c

{(\int_{a}^{b} cdx)}^{2} = \int_{a}^{b} c^{2} dx \Rightarrow (c {(b - a)}^{2} = c^{2} (b - a) \Rightarrow c^{2} {(b - a)}^{2} = c^{2} (b - a)

so if a \neq b \Rightarrow c = 0 so f = 0