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 \quad \text { L.F. }=e^{-x} $
$f(x) e^{-x}=x+C$
$f(0)=0 \quad \Rightarrow C=0 $
$f(x)=x e^x $
$A=\left|\int\limits_{-\infty}^0 x e^x d x\right|=\left.x e^x\right|_{-x} ^0-e^x \int_{-x}^p=0-(1)=1$