Generalized mean value theorem

[GeniusAreMade]

I did everything I could but couldn't find a solution to this problem using the generalized mean value theorem
How I am supposed to solve this problem, Thanks for your time

 

[Deleted member 4993]

View attachment 22383
I did everything I could but couldn't find a solution to this problem using the generalized mean value theorem
How I am supposed to solve this problem, Thanks for your time

You did not share any work yet! So we will start with the knowledge of definitions: what is the definition of "generalized mean value theorem?

Next question would be:

How would you use that definition for this problem?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[GeniusAreMade]

You did not share any work yet! So we will start with the knowledge of definitions: what is the definition of "generalized mean value theorem?

Next question would be:

How would you use that definition for this problem?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

Generalized mean value theorem: (f'(c)/g'(c))=(f(b)-f(a))/(g(b)-g(a))
I have tried to define f(x)=ln(1+x) and g(x)=arcsin(x), then I did their derivatives f' and g' and said that c is between x and x+1
But nothing worked with me, i need to solve this inequality with the generalized mean value theorem.

 

[Dr.Peterson]

Did you state the entire problem? Are there no conditions?

I graphed the three parts of the inequality, just to check that it is true, and it is true only for 0<=x<=1. For example, if x=-1/2, they are claiming that sqrt(3) = 1.732< ln(1/2)/(-pi/6) = 1.324 < 1, which is false.

Similarly, your statement of the generalized mean value theorem is incomplete; it doesn't say that this is true for all a, b, and c! It is about the existence of such a c.

Now, I find it interesting that you are to prove that an inequality is true for all x in some domain, from an existence statement (an equation that is true for some c). So my first thought might be to ponder how the two can be related at all, and hope to use that to suggest what to use for the a, b, and c in the theorem. (I haven't done so yet, as I am not yet sure I understand the whole problem.)

 

[GeniusAreMade]

Thank you for your time, I sent you all the information I have, so if you think you can solve the problem please give it a try, I've been at it for more than 5hours.
Thanks again.

 

[Dr.Peterson]

Thank you for your time, I sent you all the information I have, so if you think you can solve the problem please give it a try, I've been at it for more than 5hours.
Thanks again.

So you're saying you were asked to prove something that is not true? Then you can't do it.

I've shown that it is false unless it has a restriction to 0<x<=1:


As for a proof, my best idea so far is to try to show that if the inequality is false for a given x, it would contradict the existence of c for some interval.

But it's certainly interesting that (if x>0) the GMVT says that there is some c in (0,x) such that [MATH]\sqrt{\frac{1-c}{1+c}}=\frac{\ln(1+x)}{\arctan(x)}[/MATH].

 

[GeniusAreMade]

Thank you very much, now the ideas are clearer in my head thanks to the graph, I got what you are saying.
The doctor who gave us this problem will solve in his next lecture (in a week) I will send you the solution if he solves it.
Thanks for your help I'm very grateful.

 

[Deleted member 4993]

Thank you very much, now the ideas are clearer in my head thanks to the graph, I got what you are saying.
The doctor who gave us this problem will solve in his next lecture (in a week) I will send you the solution if he solves it.
Thanks for your help I'm very grateful.

Only other theorem involving "inequality" that I can think of is Rolle's Theorem.