critical point

Imagine walking along a hilly trail and tracking your altitude. The moments worth noticing are the spots where the ground momentarily stops climbing or falling — the very top of a rise, the bottom of a dip, or a sharp ledge where the slope changes abruptly. Those special spots are the critical points. Everywhere else the ground is clearly going up or clearly going down, so nothing interesting can be hiding there.

Precisely, a critical point of a function f is a value x in its domain where either f'(x) = 0 (the tangent line is flat) or f'(x) does not exist (a corner, cusp, or vertical tangent). The key fact, called Fermat's theorem, is that any local maximum or minimum that sits inside an interval must occur at a critical point. So critical points are the complete shortlist of candidates for the highest and lowest spots inside the interval.

A common mistake is to assume that every critical point is a peak or a valley — it is not. The function f(x) = x^3 has f'(0) = 0, yet x = 0 is neither a max nor a min; the curve just flattens for an instant and keeps rising. A critical point only flags a place where an extreme could be; you still have to test it to find out what is really going on there.