Q35 / 90 · KVPY 2014 (SB/SX) · Mathematics

A continuous function $f: R \rightarrow R$ satisfies the equation $f(x)=x+\int_{0}^{x} f(t) d t$. Which of the following options is true ?

  • A
    $f(x+y)=f(x)+f(y)$
  • B
    $f(x+y)=f(x) f(y)$
  • Correct answer
    C
    $f(x+y)=f(x)+f(y)+f(x) f(y)$
  • D
    $f(x+y)=f(x y)$

Solution

$f^{\prime}(x)=1+f(x) \quad f(0)=0$ $\frac{f(x)}{1+f(x)}=1$ $\ln (1+f(x))=x$ then $f(x+y)=f(x)+f(y) f(x) f(y)$

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

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