Injective function for a given 'a' 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 ?

$$f(x)=f(y) \Rightarrow x^3+ax^2+3x+100=y^3+ay^2+3y+100$$
$$(x^3-y^3)+a(x^2-y^2)+3(x-y)=0$$
$$(x-y)(x^2+xy+y^2+ax+ay+3)=0$$
Then $x-y=0$ and $x^2+xy+y^2+ax+ay+3 \not =0$
$$f(x)=x^2+x(y+a)+y^2+ay+3\not =0$$
$$D=(y+a)^2-4(y^2+ay+3)<0$$
$$-3y^2-2ay+a^2-12<0$$
$$3y^2+2ay-a^2+12>0$$
$$\frac D4=a^2+3(a^2-12)<0$$
$$4a^2-36<0$$
$$a^2<9 \Rightarrow a\in(-3;3)$$
