I'm taking calculus one and I have to determine if this statement is true or false.
A relative maximum or minimum must occur at a critical point.
I believe it is false. The answer key says it is true so I am curious if I am right (low probability) or if I can get this clarified (the answer key has always been right when I thought it was wrong).
For example for y = $\frac{1}{x}$ there is a critical point at x=0 (because the derivative at this point does not exist) even though there is no relative maximum or minimum (or is there?).
I think if the statement was "A relative maximum or minimum must occur at a critical point when the derivative at that point exists."
Please help. Thanks.
Best Answer
It depends on what definition of "critical point" you are using. If you mean the same thing as "stationary point" (i.e., a point where the derivative exists and equals zero), then you are right, but not for the reason that you give (see below). However, some people use the phrase "critical point" also for a point where the function $f$ is defined, but not its derivative $f'$ (including endpoints of an interval in the domain $D_f$), and if this is the definition used in your course, then you are wrong.
In your example, $f(x)=1/x$ is undefined at $x=0$, so it doesn't even make sense to talk about maximum or minimum there.
Consider instead the absolute value function $f(x)=|x|$, which has a minimum at $x=0$ despite the fact that $x=0$ is not a stationary point ($f'(0)$ doesn't exist).
[On the other hand, if $f'(a)$ exists and $f$ has a local max/min at $a$, then $a$ must be a stationary point (i.e., $f'(a)=0$).]