Extreme values of a function without calculus

Question

Given a function $f:[1,+\infty)\rightarrow\mathbb{R}$ with $f(x)=\frac{6x^2+x}{x^3+x^2+x+1}$ How do we determine whether the function has a maxima or a minima without using derivatives. It is easy to do it for quadratics since we can derive an inequality with $0$ on one side, however I am stuck when it comes to rational functions.

Answer 1

You know that $f(1)=\frac74$ and that $\lim_{x\to\infty}f(x)=0$. So, take $M\in(1,+\infty)$ such that $f(x)<\frac74$ when $x>M$ and then the point $x_0$ of $[1,M]$ at which $f|_{[1,M]}$ attains its maximum is that point of $[1,\infty)$ at which $f$ attains its maximum.