Find $n \in \mathbb{Z}$ that maximizes $ (n+10)/(n^3+10)$ $(n+10)/(n^3+10)=(n+10)/(n^3+1000-990)$
I have thought about something like this. But am not sure about how to proceed with this. So I would like some help.
 A: You have that for $n=1$ that $11/11=1$ so you want to find $$\frac{n+10}{n^3+10}> 1$$
Otherwise the maximum value is $1$.
If $n^3+10>0$ we can multiply both sides of inequality to get
$$n+10> n^3+10\\n> n^3\\0> n(n-1)(n+1)$$
This happens when $n< -1$ or $0< n< 1$ The only integer satisfying $n^3+10>0$ and $n< -1$ is $-2$ and there are no integers satisfying $0<n<1$. For $n=-2$ we get $8/2=4$ so it's greater.
Now if $n^3+10<0$ then when we multiply we must also flip signs so
$$n+10<n^3+10\\n(n-1)(n+1)>0 $$
This happens when $-1<n<0$ or $n>1$ none of those satisfy $n^3+10<0$ hence for $n=-2$ is the maximum.
A: Let $f(n)=\dfrac{n+10}{n^3+10}$.
After having asked my computer to produce a certain number of values of $f(n)$, I was convinced that 

The maximum value of $f(n)$ is $4$ and occurs for $n=-2$.

(and uniquely for this value).
Now, the proof. My idea has been to write $f(n)=4+g(n)$ and to attempt to establish that $g(n)<0$ in general with the exceptional case $g(-2)=0$.
$$g(n)=f(n)-4=\dfrac{-4n^3+n-30}{n^3+10}$$
$g(n)$ has a numerator that, rather naturaly, can be factorized with an $(n+2)$ factor (remember that $g(n)$ must be $0$ for $n=-2$), under the form :
$$-4n^3+n-30=(n+2)(-4n^2+8n-15)$$
Writing :
$$g(n)=\dfrac{n+2}{n^3+10}\underbrace{(-4n^2+8n-15)}_{h(n)}. \tag{1}$$ 
with 
$h(n)=-4n^2+8n-15=-11-4(n-1)^2$ always $<0$,
our objective will be reached if (refer to (1)), we can prove that:
$$\forall \ n \in \mathbb{Z}, \ \ \ \dfrac{n+2}{n^3+10} \geq 0 \tag{2}$$
(2) is easy to establish : let us separate :


*

*case $n=-2$ for which we get $0$, (yielding the maximal case),

*cases where $n > -2$ (for which (2) is true as quotient of two positive numbers) and 

*cases where $n<-2$ for which we can say :
$$n \leq -3 \ \implies  \ n+2\leq -1 \ \ \text{and} \ \ n^3+10\leq-17$$
thus, we obtain two negative numbers, yielding a positive quotient in (2).
