Question

Let $f(x)$ be a differentiable function such that $f(x+y)=e^{x}f(y)+e^{y}f(x)\forall x,y$ and $f^{\prime }(0)=1$. Then the area bounded by the curve $y=f(x)$ and $x$-axis, is

• A. 1
• B. $\frac{1}{2}$
• C. 2
• D. $e$

Answer 1

Answer: (A)

$f^{\prime }(x+y)=e^{x}f(y)+e^y{f}^{\prime }(x)$
$\text{ Put }x=0$
$f^{\prime }(y)=f^{\prime }(y)+e^y$
$\therefore f^{\prime }(x)-f(x)=e^x\text{ L.F. }=e^{-x}$
$f(x)e^{-x}=x+C$
$f(0)=0\Rightarrow C=0$
$f(x)=x{e}^x$
$A=\left|\underset{-\infty }{\overset{0}{\int }}x{e}^{x}dx\right|={x{e}^x|}_{-x}^0-e^x{\int }_{-x}^p=0-(1)=1$