What should be used?

Calculus Level 4

Let f ⁣:RRf\colon\mathbb R \to \mathbb R be a continuous function satisfying f(x)+0xtf(t)dt+x2=0\displaystyle f(x) + \int_0^x t f(t) \, dt + x^2 = 0 for all real xx.

Which of the following is true?


Source: KVPY 2015 (SX/SB stream).

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