For which values of $a$, $f$ is bijective?

Question

I have troubles with the following problem.

For which values of $a$ the function $f(x)=\frac{1}{3}x^3+\frac{5}{3}x^2+ax+2$ is bijective?

I tried with the injective part in the classic form (I supposed that $f(x)=f(y) $ for $x,y\in\mathbb{R}$...) but I did not get nothing.

Can someone give me a hint (not the answer, please)

Thanks for advance.

Answer 1

Note that $f'(x)=x^2+\frac{10}3x+a$. Which values of $a$ will make $f$ strictly increasing?