Question

Why can there be no relative extrema in a closed interval?

Answer 1

I really appreciate any help to understand this, because I really can't get why there can be no relative extrema at the endpoints |:

For instance, within the closed interval [2.40, 4.72] of the function \(f(x) = sin(x) + cos(4x)\), the relative extrema lie at the endpoints, while the absolute extrema lie somewhere in between:

http://s14.postimg.org/v6aeq2vup/Relative_Min_Max.jpg

If the reason is because I can't know for sure whether the function is differentiable coming from the side left out of the interval (to say, coming from the right toward the rightmost endpoint), then how can a function like the of the image below have a relative maximum at x = -2, when the derivative at that point doesn't exist?

http://s30.postimg.org/pt0h2q61t/Relative_Min_Max2.jpg

I know that there can be a critical point when f' = 0 or undefined, but I'm trying to point out that having relative extrema only in open intervals seems a somewhat arbitrary definition, not linked with any of the thing about critical points I've learned so far ):

Thanks !!

Answer 2

@uri

Answer 3

Sure there can be relative extrema in a closed interval. But it does not make sense to talk about relative extrema at the endpoints because you don't know what happens outside of the interval.

For example, x and |x| look the same on [0, 1] but only one of them has a relative extremum at the origin. If you only know what happens on [0, 1] you can't tell them apart. We need to restrict ourselves to avoid ambiguity.

Answer 4

Relative extrema and absolute extrema are the concepts indicate to the high points and highest points in the interval. What makes you think absolute extrema must bound relative one?