Critical Numbers question?

Question

If it asks you to find critical numbers is it [a,b] or (a,b)

Is only (a,b) used for rolle's theorem and mean value theorem?

So if it gives you a function 4/x +x [1,2] find the critical numbers you count 2 as a critical number? If the same equation is used for rolle' theorem you dont count 2?

Answer 1

Critical numbers are x values of the derivative that are zeros or undefined at that point. The only case in which you would test the end points is if they were asking for the max and min and there were no critical numbers within the interval. In that case, the endpoint(s) would be the max and min values.

Answer 2

If the question uses square brackets, then you must analyse 2 to see if it is a critical point.

D[4/x +x] = -1(4/x^2) + 1
At x=2, D is zero, therefore x=2 is a critical point.
Next you must see if 2 is part of the domain. The domain is given as [1, 2], therefore 2 is part of the domain.

If you had been given a domain of [1, 2) or (1, 2), then 2 is NOT in the domain and would not have counted.

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In Rolle's theorem, the interval is usually meant to be the interval between the two points where the curve crosses the x-axis. To 'cross' the x-axis implies that the slope is not 0 at the crossing point. Therefore, it is useless to include the crossing points in the interval of interest. That is why we use (a, b).

If you were to solve Rolle's theorem using a closed interval [a, b], you should be able to show that the point where the slope is zero cannot be a nor b (the crossing points). So might as well exclude them right from the start and use (a, b) instead of [a, b].

Answer 3

For a continuous function f(x) defined over a closed interval
[a, b] it's possible for f(a) and f(b) to be a max or min

For Rolle's theorem (a special case of the mean value theorem), I think the function HAS to be defined over the closed interval (otherwise you don't have an f(a) and f(b) to work with. However, the derivative of f(x) evaluated at x = c, where c is between a and b produces the same slope as a straight line between the endpoints of the function, f(a) and f(b).

For your particular problem
function 4/x +x [1,2]

[f(b) - f(a)] / [b - a] correspond to [4-5] / [2-1] = -1

The c value that makes the derivative of your function be the same -1 is f ' (sqrt 2)
That is, when you evaluate the derivative of your function at sqrt 2, you'll get -1. [And sqrt 2 is certainly in the interval of your domain.]

Answer 4

When finding Critical Numbers on a closed interval you need to find the values at the endpoints as well - 1 and 2 will both be extrema if not critical numbers.

Rolle's Theorem says that f(x) must be continuous over the interval [1,2], but it only has to be differentiable over (1,2).

Answer 5

Whats the answer to your problem?

Answer 6

huh?