Q47 / 90 · KVPY 2015 (SB/SX) · Mathematics

Let $f: R \rightarrow R$ be a continuous function satisfying $f(x)+\int_{0}^{x} t f(t) d t+x^{2}=0$ for all $x \in R$. Then

  • A
    $\lim _{x \rightarrow \infty} f(x)=2$
  • Correct answer
    B
    $\lim _{x \rightarrow-\infty} f(x)=-2$
  • C
    $f(x)$ has more than one point in common with the $x$-axis
  • D
    $f(x)$ is an odd function

Solution

$f^{\prime}(x)+x f(x)+2 x=0$ $\frac{d y}{d x}+x y=-2 x$ I.F. $=e^{x^{2} / 2}$ $\Rightarrow y \cdot e^{x^{2} / 2}=\int e^{x^{2} / 2}(-2 x) d x+C$ $e^{x^{2} / 2} \cdot y=-2 e^{x^{2} / 2}+C$ $y=-2+C e^{i x^{2} / 2}$ $f(0)=0 \Rightarrow C=2$ $f(x)=2\left[e^{-x^{2} / 2}-1\right]$

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