Local maximums/minimums

Critical points and candidates

The possible locations of local maxima or minima? Your book says:

If a function, continous on an interval (its domain), has a local maximum or minimum at $p$, then $(p,f(p))$ is a critical point or an endpoint of the interval.

But not every critical point or endpoint is a local max or min.

Consider this function (depicted graphically) with a domain of $x_1\leq x \leq x_{10}$ (except for $x_3$, since $f(x_3)$ is not defined):

Critical points, $(p, f(p))$, are points where

  • $f(p)$ is in the domain of $f$.
  • $f'(p)=0$ or $f'(p)$ is undefined.

Open question: is $f'(p)$ defined at an endpoint of an interval on which $f$ is continuous? or not?

We can debate this, but with the definitions above, whether the derivative is defined or not at the endpoint doesn't matter: It's a candidate because it's an endpoint of the domain.