Question

Let $f\left(x\right)$ be a derivable function, $f^{\prime }\left(x\right)>f\left(x\right)$ and $f\left(0\right)=0$. Then

• A. $f\left(x\right)>0$ for all $x>0$
• B. $f\left(x\right)<0$ for all $x>0$
• C. No sign of $f\left(x\right)$ can be ascertained
• D. $f\left(x\right)$ is a constant function

Answer 1

The correct option is A $f\left(x\right)>0$ for all $x>0$

$f^{\prime }\left(x\right)-f\left(x\right)>0$
$\frac{df\left(x\right)}{dx}-f\left(x\right)>0$

By using integrating factor $e^{-x}$
$e^{-x}\frac{df\left(x\right)}{dx}-e^{-x}f\left(x\right)>0$
$\Rightarrow \frac{d\left(e^{-x}f\right(x\left)\right)}{dx}>0$
$\Rightarrow e^{-x}f\left(x\right)$ is increasing function
$\Rightarrow e^{x}f\left(x\right)>e^{-0}f\left(0\right)\forall x>0$
$\Rightarrow f\left(x\right)>0\forall x>0$