Question

Question about relative minima and maxima.

Answer 1

The definitions of relative extrema are:

\(f(c)\) is a relative maximum of \(c\) if there exists an interval \((a,b)\) containing \(c\) such that \(f(c)\geq f(x)\) for all x in \((a,b)\).

\(f(c)\) is a relative minimum of \(c\) if there exists an interval \((a,b)\) containing \(c\) such that \(f(c) \leq f(x)\) for all x in \((a,b)\).

Why can't it be an interval \(\large{[a, b]}\)?

Answer 2

that is a good question

Answer 3

relative max means relative to what is around it in both directions, right and left
that is why open intervals and not closed

Answer 4

Does it mean that when we talk about relative extrema, we are considering the whole domain of f(x)?

Answer 5

@satellite73

Answer 6

This video explanation is quite good and may help you: https://www.khanacademy.org/math/differential-calculus/derivative_applications/absolute-relative-maxima-minima/v/relative-minima-maxima

Answer 7

I'm having a hard time understanding this /:

In fact my doubts raised after watching the very same Khanacademy videos.

Answer 8

(I've earned a lot of point rewatching that video and I still don't get it ): )

Answer 9

\[\large (a, b) \subset [a, b]\]

Clearly if you can find a closed interval with the given requirement, it works!

We use open interval in definition precicely because open interval is enough for testing extrema. Closed interval is NOT needed. When you define something, you want conditions to be as less restrictive as possible.

Answer 10

Then, what is the reason why we need to be more restrictive when finding absolute extrema? (I'm recalling the Extreme Value Theorem).

Answer 11

Think of the "definition" of relative extrema - e.g. x=a is a relative minimum if values "just around x=a" are greater than the value at x=a.

If you INCLUDED the end points then you cannot look at values "around" those points

Answer 12

Do you prefer saying 1) \( x\ge 1 \implies x^2 \ge 1 \) or 2) \(x\ge 1 \implies x^2 \ge 0\) ?

Do you see how both statements are similar/different ?

Answer 13

you become "restricted" to looking at only values to "one-side" of those end points

Answer 14

I /believe/ that "relative" extrema's have to be u-shaped or n-shaped where-as "absolute" extrema do not

Answer 15

Let suppose that my function f(x) is equal to 3 at x = 2 and it's a relative maximum. If I have a closed interval [1.5, 2.5], with the derivative, I should be able to show that f(1.5) and f(2.5) are \(\leq 3\).

Answer 16

What I don't understand when finding that relative maximum is that my interval must be (1.5, 2.5) instead of what I wrote above ):

Answer 17

Surely something I'm misunderstanding, because I can't see why a closed interval can be valid there.

Answer 18

Consider only the open interval then ?

Answer 19

if it is a relative extrema in open interval then it will also be a relative extrema in closed interval

Answer 20

Consider this curve: