What is the SOP for proving inequalities by Mean Value Theorem? For example, for $x>0$, $1+x<e^x$. I know the proof of this is to apply MVT to $e^x$($x>0$), and one gets $e^x-e^0=e^{\xi}x$, where $\xi$ is between $0$ and $x$, then $\frac{e^x-1}{x}=e^{\xi}>1$, hence $1+x<e^x$ for all $x>0$. However, I always feel frustrated when doing such problem. First, how do we know that we should apply MVT to the function $e^x$? By what clues or signs? I had tried to apply MVT at $F(x)=e^x-(1+x)$, but can't deduce any meaningful things. Second, how do we know that we should pick the reference point at $0$? Also, I don't know how to solve $x>\ln(1+x)$ for all $x>0$. Do there exist a SOP(Standard Operating Procedures) or thinking routine to such questions? 
And on the other hand, how do you know that such questions are NOT related to the usage of Taylor expansions(I mean for order higher than $1$), while some questions must use Taylor expansions? For me, when facing the question $1+x<e^x$, I may also want to try the way 
 rewriting $e^x=1+x+\frac{1}{2}x^2+\frac{e^{\xi}}{3!}x^3$ as Taylor expansion, but this appears not help to solve the question. However, let's see another problem, proving $1+(1/2)x-(1/8)x^2<\sqrt{1+x}$ for all $x>0$ requires the Taylor expansions(but not MVT this time!!). How to distinguish them? (Also note that proving $\sin(x) \ge x-\frac{x^3}{6}$ requires MVT but not Taylor...) These questions frustrated me a lot. Need your help. 
 A: *

*Why do we pick $0$ as a reference point in the initial proof?


We have $e^0=1$, so it makes sense to work with this point, as $1$ appears in the desired inequality. What you're really being asked to show here is that $e^0+x<e^x$. Noticing that is the trick. If you haven't seen many arguments like this before, it's less likely you would notice it. Once you decide, "perhaps I'll try MVT", it makes sense to try and see values in the problem as outputs of some function involved. As for why $e^x$ and not $e^x-(1+x)$, it's a good principle to start with something simple, and only try something complicated if the simple thing doesn't work. Additionally, the denominator from the MVT plays a role. If you tie up the entire inequality in $F$, then there's nothing left to go "downstairs".


*How to show $x>\ln(x+1)$ for $x>0$?


This is simply the inequality you already have, with logs taken on both sides. It is valid to apply $\ln$ to both sides of an inequality, because $f(x)=\ln(x)$ is a strictly increasing function, so it preserves order relations.


*Why not use Taylor series?


Writing $e^x=1+x+(\text{positive terms})$ certainly does work to prove the inequality in question, very elegantly. It is often the case that multiple methods are effective, and it might be possible to show a fact using Taylor series or using MVT. Which one you choose is just a matter of personal preference.


*How to choose MVT vs. Taylor in general?


I'm not sure there's a satisfying way to answer this question in general. Some people might find an MVT proof more sleek and elegant than a Taylor expansion proof, but you can only use MVT if you can find a way to line up the pieces appropriately. This isn't always obvious or easy to do. In your example with the square root function, the messy expression on the left seems to be a challenge to incorporate into the relatively sparse MVT format. As for the example with the sin function, I'd be inclined to use Taylor expansions to show that. Are you sure it can't be done that way?
A: I would just say experience in having proved a number of these.
Also, the inequality involving
$\sqrt{1+x}$
can be proved just by squaring both sides.
