Question

Describe as precisely thoroughly as possible why critical values of a function f(x) occur either where f′(x) = 0 or where f′(x) is undefined.

Answer 1

Well, most books define critical values of a function as x-values where f'(x)=0 or f'(x) is undefined, so they occur there because that's the definition of what a critical value is.

But the reason we define it that way is that when you're looking for local extrema (local mins and/or local maxs), these must occur when f'(x)=0 or f'(x) is undefined.

If f has a local extremum at a point x=c, then f'(c)=0 or f'(c) is undefined.