Find values of $a$ if $f$ is a one-one function 
If $$f(x) = x^3 + (a+2)x^2 + 5ax + 5$$ is a one-one function then find the set of values of $a$.

I know that I need to find values of $a$ for which function is strictly monotonic increasing or strictly monotonic decreasing and we check monotonicity of function by its derivative $f'(x)$.
here $$f'(x) = 3x^2 + 2x(a+2) + 5a$$
so, what condition should I impose on $f'(x)$ to make the given funtion one one or strictly monotonic?
 A: HINT. You know that $f'(x)$ is a parabola. To make $f(x)$ always increasing, you want $f'(x)>0$; that is, you want the parabola to always be above the $x$-axis. It touches the $x$-axis when it only has one root. What formula can you use to see what makes this only have one root in terms of $a$? Would this then give you a condition on $a$ to force the parabola to be always above the $x$-axis?
A: A differentiable function is strictly monotonic when its derivative is not $0$ on any segment, meaning if $f'(x)=0$ then there is a small $z>0$ with the property that for any $0<y<z :f'(x+y)\neq 0,f'(x-y)\neq 0$, and if $\forall x:f'(x)\geq 0$ or $\forall x:f'(x)\leq 0$.
Try and use that. You are on the right track.
A: Consider $y=ax^2+bx+c,$ where $a>0.$ Now $$y=a(x^2+\frac{b}{a}x+\frac{c}{a})=a\big((x+\frac{b}{2a})^2+\frac{c}{a}-\frac{b^2}{4a^2}\big)=a\big((x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a^2}\big).$$ So if you have $4ac-b^2\geq0$ then $y$ will be non-negative for all $x.$ 
Try to connect these ideas to your problem. 
A: Note that $\lim_\limits{x\to -\infty} f(x)=-\infty$ and $\lim_\limits{x\to +\infty} f(x)=+\infty$. That is why you need to make sure the function $f(x)$ is strictly increasing $\left(x_1<x_2 \Rightarrow f(x_1)<f(x_2)\right)$. It implies:
$$f'(x)\ge 0 \Rightarrow 3x^2 + 2x(a+2) + 5a\ge 0 \Rightarrow D\le 0 \Rightarrow (a+2)^2-15a\le0 \Rightarrow \\
\frac12 \left(11 - \sqrt{105}\right)\le a\le\frac12 \left(11 + \sqrt{105}\right).$$
See WA animation for $f'(x)$.
Also note that $f'(x)=0$ when $a=\frac12 \left(11 - \sqrt{105}\right) \ \left(\text{or} \ a=\frac12 \left(11 + \sqrt{105}\right) \right)$ and $x = \frac{\sqrt{35/3}}{2} - \frac52 \ \left(\text{or} \ x = -\frac52 - \frac{\sqrt{35/3}}{2}\right)$, which implies it is an inflection point (which is still one-to-one).
See WA animation for $f(x)$.
