Question

$y=f\left(x\right)$ is a function which satisfies (i)$f\left(0\right)=0$
(ii) $f^{\prime \prime }\left(x\right)=f^{\prime }\left(x\right)$ and (iii) $f^{\prime }\left(0\right)=1$
then the area bounded by the graph of $y=f\left(x\right)$ the lines $x=0,x-1=0$ and $y+1=0$ is

• A. $e$
• B. $e-2$
• C. $e-1$
• D. $e+1$

Answer 1

Answer: (C)

$e-1$

If $f^{\prime }\left(x\right)=x+c\Rightarrow f^{\prime }\left(x\right)=k{e}^x$
$\Rightarrow f^{\prime }\left(x\right)=e^x$ $\left\{f^{\prime }\right(0)=1\Rightarrow k=1\}$
$\Rightarrow f\left(x\right)=e^x+\lambda$
$\Rightarrow f\left(x\right)=e^x-1$ $\left\{f\right(0)=0\Rightarrow \lambda =-1\}$
$A={\int }_0^1(e^x-1+1)dx=e-1$