<s>[INST] Taylor theorical question: solving an inequality Prove by using Taylor formula that in a neighbourhood of zero this inequality holds:
$\cosh x  \le 12 \frac{x^2+2}{24-x^4}$
Please explicitate everything you do :)
Thanks in advance
 [/INST] The hint pretty much tells you everything you need to do: take the Taylor expansion of the difference of the two sides, until you see a non-zero term. Then check the sign of that term near $0$.
In particular, the theorem that we will use is the following:
Theorem. Let $f,g$ be two functions defined on a (pointed) neighborhood of zero, and such that $g$ does not cancel there. Then, if $f(x) = g(x) + o(g(x))$ when $x\to 0$, $f$ and $g$ keep the same sign on a neighborhood of $0$.
Proof. We have $\frac{f(x)}{g(x)} = \frac{g(x)+o(g(x))}{g(x)} = 1+o(1) \xrightarrow[x\to 0]{} 1$, and thus there exists $\delta > 0$ such that $\frac{f(x)}{g(x)} \in \left(\frac{1}{2},\frac{3}{2}\right)$ for all $x\in(\delta,\delta)$. $\qquad\square$
Now, we will invoke this theorem at the end: we will prove that $12\cdot \frac{2+x^2}{24-x^4} - \cosh x = \alpha x^c + o(x^c)$ for some $\alpha\neq 0,c>0$ (by performing a Taylor expansion of both sides of the inequality until the point where they differ) and be quite happy to notice that $c$ is even, and $\alpha > 0$. Then, we will "set" $g(x) = \alpha x^c$ and conclude.

When $x\to 0$, we have (this one is standard)
$$
\cosh x = 1+\frac{x^2}{2}+\frac{x^4}{24}+\frac{x^6}{720}+o(x^6)
$$
and 
$$\begin{align}
12\cdot \frac{2+x^2}{24-x^4} &= 
\frac{1+\frac{x^2}{2}}{1-\frac{x^4}{24}}
= \left(1+\frac{x^2}{2}\right)\left(1+\frac{x^4}{24}+O(x^8)\right)\\
&= 1+\frac{x^2}{2}+\frac{x^4}{24}+\frac{x^6}{48}+O(x^8)\\
&= 1+\frac{x^2}{2}+\frac{x^4}{24}+\frac{x^6}{48}+o(x^6)
\end{align}$$
and thus
$$
12\cdot \frac{2+x^2}{24-x^4} - \cosh x  = \frac{7}{360}x^6 + o(x^6).
$$
Since $\frac{7}{360}>0$, the difference is positive on a neighborhood of $0$, as it is equivalent to $\frac{7}{360}x^6$. This shows that on that very same neighborhood of $0$, the inequality $$
12\cdot \frac{2+x^2}{24-x^4} > \cosh x$$ holds by the above theorem.
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