I came upon a question that asks to find a function which has a critical point at some coordinates and an inflection point at some other coordinates. I can't see exactly what is the difference between the two, could someone help clarify it to me?
Thanks!
Best Answer
Critical points refer to the first derivative. In particular, $x=a$ is a critical point of $f(x)$ if either $f'(a)=0$ or $f'(a)$ is not defined. The importance here is that all maxima or minima are found at critical points or endpoints of a domain. So a common way to find extrema (maxima and minima) is to find the endpoints and critical points and see which of those are extrema.
Inflection points refer to the second derivative. In particular, $x=a$ is an inflection point of $f(x)$ if the second derivative of $f$ is positive in an interval immediately on one side of $a$ and negative in an interval immediately on the other side of $a$. (I believe it is also a condition that $f'(x)$ exists.) It is also true that either $f''(a)=0$ or $f''(a)$ is not defined, but those conditions are not enough to guarantee an inflection point. The importance here is that $f(x)$ is concave up (turning up) on one side of $x=a$ and concave down (turning down) on the other side of $x=a$.
Both critical points and inflection points have many other uses.