Q47 / 160 · WBJEE 2015 · Mathematics

Let $f: R \rightarrow R$ be a continuous function which satisfies $f(x)=\int_{0}^{x} f(t) d t .$ Then. the value of $f\left(\log _{e} 5\right)$ is

  • Correct answer
    A
    0
  • B
    2
  • C
    5
  • D
    3

Solution

Given, $\quad f(x)=\int_{0}^{x} f(t) d t$ Using Leibnitz theorem, we get $$ f^{\prime}(x)=f(x) \Rightarrow f(x)=k e^{x} $$ On putting $x=0$, we get $f(0)=\int_{0}^{0} f(t) d t$ $\Rightarrow \quad k e^{0}=0$ $\left.\because \int_{a}^{a} f(x) d x=0\right]$ $\left.\Rightarrow \quad k=0 \quad \mid \because e^{0}=1\right]$ $\therefore \quad f(x)=0 \Rightarrow f\left(\log _{e} 5\right)=0$

Hence, the correct answer is option: 'A'.

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