Can relative maximum and minimum points exist when a function is defined at say x=c, however the derivative does not exist or tends to infinity? Ie the graph of. F (x)= |x|, for x=c=o. If I am correct the relative minimum is at o, can it also be the abs minimum? I recalled the theorem by memory. Let the function f be defined on the closed interval from a to b and have a relative maximum or minimum at x=c, where c is in the open interval from a to b. If the derivative exist as a finite number at x=c, then the derivative is o at that point. I went through the derivation of the proof, and the theorem (by analyzing my derivation), does not say what happens when. 1) when the derivative fails to exist at a point c. 2) at the endpoints, or that there is always a max or min when the derivative equals 0. I sometimes tutor my little neighbor next door for free, because it helps me practice things I havnt used in a while. Today I told him, that a relative max or min can occur if the function is defined at that point, does not have to be necessary differential at that point. I use y=|x| as an example. I felt bad, because I believed I gave him the wrong answer. Sorry for the long post.