i made a limit on both infinity and minus infinity for them

and i tried to find but its not working

http://img201.imageshack.us/img201/5458/23597303em5.gif

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- December 20th 2008, 08:26 AMtransgalactichow to find the max,min,sup,inf of these cases..
i made a limit on both infinity and minus infinity for them

and i tried to find but its not working

http://img201.imageshack.us/img201/5458/23597303em5.gif - December 20th 2008, 09:55 AMbkarpuz
The domain of the function is , with .

Let

then

which implies .

Therefore its maximum value is .

On the other hand, attains its inferior value as (as you have shown).

For the second one, let

for (I believe it must be given on the positive halfline).

Clearly, as or , we have (superior value).

And on the other hand, we have

which implies

and thus for all (minimum value).

You may similarly show the third one too, I have to say that this one is very interesting, just show that its maximum values decrease and similarly show that its minimum values increase. - December 20th 2008, 11:24 PMtransgalactic
so the max and the sup of the first one is 1

what is the min and inf of the first one? - December 21st 2008, 12:41 AMtransgalactic

the maximal number is infinty

when x->+infinty

the smallest value is found when

x->-infinity

?? - December 21st 2008, 01:09 AMtransgalactic
regarding the third one:

the derivative is

(cosx * x - sin x)/x^2

how to find the extreme points in order to find max min - December 21st 2008, 01:42 AMbkarpuz
implies which is the

**inferior**value, because there is no point that holds.

As I said before is unbounded; i.e., its**superior**value is .

And it takes its**minimum**value at with .

**Note**. To find the extreme points of a function , just find the zeros of the equation .

Say is such a point that , then if we say that has a minimum point at , and if we say that has a maximum value at .

**PS**. I just now saw that it seems hard by proving in the way I told you before. - December 21st 2008, 02:08 AMtransgalactic
in the third case the sinus function

i get

cosx * x -sinx =0

how to solve this equation?? - December 21st 2008, 02:31 AMmr fantastic
An obvious solution is x = 0. The function has a removable singularity at x = 0. It makes sense to define f(0) = 1.

Then the global maximum of is located at (0, 1).

The other local maxima (and local minima) cannot be found algebraically. You will need to use a numerical procedure to get approximate solutions. - December 21st 2008, 02:38 AMbkarpuz
Yes it is not easy, thus, we have to find other ways to find max and min values of this function.

May the the following analysis may be helpful to you.

Let for .

Clearly, for all , we have .

Thus, if there exists a point such that holds, then we can say that is the maximum value of on .

By calculating , which implies , therefore is its maximum value.

I saw**mr fantastic**'s arguments after posting the answer.