# Math Help - Relative Extrema

1. ## Relative Extrema

I don't understand this. Some resources state that endpoints cannot be relative extrema while others state they can be. So which is it?

2. Do your sources say the can't be relative extrema or that they might not be?

For example, a "relative extremum" (a relative minimum) of $f(x)= x^2$ on the interval [0, 1] is at x= 0, an end point.

Strictly speaking, I would say that x= 1 is also a relative maximum because it is larger than any nearby value of the function in that interval.

Of course, it is not a "critical point"- the derivative is not 0 at x= 1. x= 0 is both an endpoint of the interval and a critical point of the function.

3. Well I mean that the one textbook I have has endpoints as relative extrema and an older textbook (different publisher), while it doesn't explicitly state they aren't, it none the less excludes them from the answer key. So with your example x=0 and x=1 would be absolutes not relative.

I realize I'll ask the instructor what he wants on the test, but other than that I'm wondering is there any consensus on it?

4. Originally Posted by Mr.Berlin
Well I mean that the one textbook I have has endpoints as relative extrema and an older textbook (different publisher), while it doesn't explicitly state they aren't, it none the less excludes them from the answer key. So with your example x=0 and x=1 would be absolutes not relative.

I realize I'll ask the instructor what he wants on the test, but other than that I'm wondering is there any consensus on it?
A relative extremum is generally a value greater than any other in some open interval containing the point where the extrema is achieved. This automatically excludes endpoints since there is no such open interval within the range under consideration.

(Thus an end point cannot be a relative extremum even if it is a stationary point with non-zero second derivative)

CB