If a function is given whose domain and codomain are all real numbers such that
$$f(x) = x^3 + ax^2 + 3x + 100$$ then we have to find the value of $a$ for which the function is injective .
For that first I prove it as increasing or decreasing function . By doing this i got that the function is increasing when $a$ belongs to $[-3,3]$ and the function cannot be decreasing for any value of $a$.
But then I thought of second method by doing $f(x) =f(y)$. Solving this I got a equation of hyperbola i.e
$$x^2 +xy +y^2 + ax + ay +3 =0$$
But now how to proceed ?