What should be used?

Calculus Level 4

Let $f\colon\mathbb R \to \mathbb R$ be a continuous function satisfying $\displaystyle f(x) + \int_0^x t f(t) \, dt + x^2 = 0$ for all real $x$ .

Which of the following is true?

Source: KVPY 2015 (SX/SB stream).

f( x )

has more than one point in common with the

x

-axis

\lim _{x \to \infty}f(x)=-2

f( x )

is an odd function

\lim _{x \to \infty}f(x)=2