0-x-f-t-dt-x-2x-tf-t-dt-0-2x-1-t-f-t-dt-f-1-2-

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Question Number 329 by 123456 last updated on 23/Dec/14

$$\underset{0}{\stackrel{x}{\int }}f\left(t\right)dt+\underset{x}{\stackrel{2x}{\int }}tf\left(t\right)dt=\underset{0}{\stackrel{2x}{\int }}(1-t)f\left(t\right)dt$$$$f\left(\frac{1}{2}\right)=?$$

Commented by prakash jain last updated on 30/Dec/14

$$\mathrm{differentiate}\mathrm{w}.\mathrm{r}.\mathrm{t}x$$$$f\left(x\right)+2xf\left(2x\right)\frac{d}{dx}2x-xf\left(x\right)=(1-2x)f\left(2x\right)\frac{d}{dx}2x$$$$f\left(x\right)+4xf\left(2x\right)-xf\left(x\right)=2(1-2x)f\left(2x\right)$$$$x=\frac{1}{2},f\left(\frac{1}{2}\right)+2f\left(1\right)-\frac{1}{2}f\left(\frac{1}{2}\right)=0$$$$f\left(\frac{1}{2}\right)=-4f\left(1\right)$$

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