Question

Let $f:R\to R$ a continuous function which satisfies $f\left(x\right)={\int }_0^{x}f\left(t\right)dt$. Then the value of $f\left({\log }_{e}5\right)$ is

• A. $0$ (zero)
• B. $2$
• C. $5$
• D. $3$

Answer 1

The correct option is A $0$ (zero)

$f^{\prime }\left(x\right)=f\left(x\right)$ (Leibnitz theorem)
$\Rightarrow f\left(x\right)=k{e}^x$ and $f\left(0\right)={\int }_0^{0}f\left(t\right)dt=0\Rightarrow k=0\therefore f\left(x\right)=0\Rightarrow f\left({\log }_{e}5\right)=0$