April 2nd, 2018, 10:25 PM  #1 
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  How can I show that f(a) = f(b)?
Show that if f(x) is increasing and concave up on [a,b], then f((a+b)/2) < (f(a) + f(a))/2. My attempt: I would use the Mean Value Theorem. Since the function is increasing and concave up on the given interval, f((a+b)/2) < f(b). By the Mean Value Theorm, there exists a value c in the interval such that f ' (c) = (f(b)  f(a))/(ba) > 0 since the function is increasing. That means, f (a) < f(b). So, f((a+b)/2) < f(b) = f(b)/2 + f(b)/2. I'm stuck. Since f(a) < f(b), I don't know how to complete the proof. Please help me. Thanks a lot. 
April 2nd, 2018, 10:40 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,343 Thanks: 732  
April 3rd, 2018, 04:29 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
I presume you mean that f((a+b)/2)< (f(a)+ f(b))/2. You assume, without saying it explicitly, that f is differentiable and that b> a. Are those given as part of the problem? 
April 3rd, 2018, 12:48 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,788 Thanks: 708 
The inequality comes from the definition of concave upward. You need to use a mathematical definition of this term.

April 3rd, 2018, 06:51 PM  #5  
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
https://en.wikipedia.org/wiki/Convex_function  
April 5th, 2018, 10:00 AM  #6  
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  Quote:
 
April 5th, 2018, 11:41 AM  #7 
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
Proved what? That every convex function is convex?

April 5th, 2018, 01:50 PM  #8 
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  "it" refers to the inequality as stated in the hypothesis. That is, if ..... then prove that f((a+b)/2) < f(a)/2 + f(b)/2. One of my club members proved this hypothesis elegantly. He won't be able to attend the club for two months because he'll be on vacation. Anyways, I'll ask him for his proof when I see him again. I was just eager to know how a proof would be done for this one. I guess I'll just wait for him to come back. I won't talk about this post anymore. I will leave it right here. 

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