I know that the second derivative should be checked for determining whether the point of inflection is maxima or minima. However, in many textbooks, websit's, research papers, it is usually ignored. Are there any conditions for that to hold true or is it getting ignored for being obvious? It would be great if someone can answer this?

I don't think there is any kind of clear rule as to when this is obvious. Sometimes in problems that have a physical interpretation, it is clear that there can be no local maxima, and so the answer must be a minimum, for example calculating the equilibrium distance of a planet from the sun given the planet's angular momentum. Another example is the standard calculus text problem of finding the height of a square box which will enclose the greatest volume for a given area of material in the walls. It's geometrically obvious that very tall or very short boxes have less volume, so if there is only one height for which the derivative is zero, then it must be a maximum.

So in some contexts, where there is a clear physical or geometric interpretation, you can skip checking the second derivative altogether. In other cases, such as research papers, it may simply be assumed that you would check and that the test was passed, or you wouldn't be publishing anything. In other words, in some circles it is considered so basic that it is not worth mentioning, like saying "I put on my socks before my shoes, not after" -- everyone knows how it goes and simply assumes that you did it right.

--Stuart Anderson