How important are minima, maxima and point of inflection in differential calculus?

Question

I've recently started working through an introductory book on calculus. Having done the greatest portion of differential calculus prior to reading the book, I would like to try and understand how important the role of minima, maxima and points of inflection are.

I understand that they are very useful when identifying how to sketch a graph, and of course I understand how they tie in with derivatives and differentiation, but do they have any immediately prominent implications?

If the question is too vague please notify me and I will adjust my wording before closing the question please, thank you.

Answer 1

Your question is not vague, but it is general. And so will this answer be.

Understanding maxima and minima, how to find them is extremely important (and very easy, once your understand the intuition of the derivative as the slope of a curve). I have had a long career in scientific research and then in finance. I can't imagine making do without them.

So, yes. You should get some intuition around what the first and second derivatives are (just chat with someone who understands and they will draw you a curve and explain). Once you get that intuition, the intuition behind points of inflection and how to find them will follow naturally.

I have to say, in my career in applied math, I have rarely had to make use of points of inflection, though. Still, I can't imagine any of my colleagues at any state in my career who didn't have a solid grasp of that too.

Answer 2

Studying applications to curve-sketching is essential to understanding more theoretical material that will probably follow later in the course so I would make sure you have a good practical (and not only theoretical) background with solving concrete problems using basic facts on derivatives. In my experience, students who don't "get their hands dirty" with this type of exercise have difficulty later on with separting the wheat from the chaff when they need to solve more advanced problems and often pursue completely hopeless avenues of solution due to inexperience with basic principles.

Historically speaking, work with maxima, minima, tangents, inflection points, etc. in the 17th century by such greats as Pierre Fermat, Gottfried Leibniz, and others is what led later to more advanced developments in analysis. The source of analysis is geometry--and, yes, curve sketching.