Question

Let $f:[0,\infty )\to R$ be a continuous function such that $f\left(x\right)=1-2x+{\int }_0^x{e}^{x-t}f\left(t\right)dt$ for all x$ϵ[0,\infty )$. Then, which of the following statement(s) is (are) TRUE?

• A. The curve $y=f\left(x\right)$ passes through the point $(1,2)$
• B. The curve $y=f\left(x\right)$ passes through the point $(2,-1)$
• C. The area of the region $\left\{\right(x,y\left)ϵ\right[0,1]\times R:f(x)\le y\le \sqrt{1-x^2}\}$ is $\frac{\pi -2}{4}$
• D. The area of the region $\left\{\right(x,y\left)ϵ\right[0,1]\times R:f(x)\le y\le \sqrt{1-x^2}\}$ is $\frac{\pi -1}{4}$

Answer 1

Answer: (B), (C)

The curve $y=f\left(x\right)$ passes through the point $(2,-1)$
The area of the region $\left\{\right(x,y\left)ϵ\right[0,1]\times R:f(x)\le y\le \sqrt{1-x^2}\}$ is $\frac{\pi -2}{4}$

$f\left(x\right)=1-2x+{\int }_o^x{e}^{x-t}f\left(t\right)dt$
$\Rightarrow e^{-x}f\left(x\right)=e^{-x}(1-2x)+{\int }_o^x{e}^{-t}f\left(t\right)dt$
Differentiate w.r.t.x.
$-e^{-x}f\left(x\right)+e^{-x}{f}^{\prime }\left(x\right)=-e^{-x}(1-2x)+e^{-x}(-2)+e^{-x}f\left(x\right)$

$\Rightarrow -f\left(x\right)+f^{\prime }\left(x\right)=-(1-2x)-2+f\left(x\right)$
$\Rightarrow f^{\prime }\left(x\right)-2f\left(x\right)=2x-3$

Integrating factor $=e^{-2x}$
$f\left(x\right)\cdot e^{-2x}=\int e^{-2x}(2x-3)dx$
$=(2x-3)\int e^{-2x}dx-\int \left(\right(2\left)\int e^{-2x}dx\right)dx$

$=\frac{(2x-3)e^{-2x}}{-2}-\frac{e^{-2x}}{2}+c$

$f\left(x\right)=\frac{2x-3}{-2}-\frac{1}{2}+c{e}^{2x}$

$f\left(0\right)=\frac{3}{2}-\frac{1}{2}+c=1\Rightarrow c=0$

$\therefore f\left(x\right)=1-x$

Area $=\frac{\pi }{4}-\frac{1}{2}=\frac{\pi -2}{4}$.