I was reading a book on Calculus, by Michael Spivak. There they mention that points where the derivative is equal to zero are called critical points. They nowhere mention that where the derivative does not exist will be a critical point. When I Googled it, they say critical points are those where the function derivative is either zero or the derivative does not exist. Which definition should I take?
$\endgroup$ 91 Answer
$\begingroup$When the derivative is 0 at a point $(x,y)$, that point is critical. When a derivative does not exist, there might be no single point that can be labeled as critical. For example, the function $x, x\in (-\infty, 0)$ and $x+3, x\in [0, \infty)$. The derivative does not exist at $x=0$, however there is no single point that can be labeled as critical. So in my opinion, if $\lim_{x\to a} f(x) $ exists, then that is a critical point.
$\endgroup$
