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April 2nd, 2018, 10:25 PM   #1
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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))/(b-a) > 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.
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April 2nd, 2018, 10:40 PM   #2
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Quote:
Originally Posted by davedave View Post
f((a+b)/2) < (f(a) + f(a))/2.
Is that copied correctly? The rhs is f(a). Also you probably need to use the definition of concavity at some point.
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April 3rd, 2018, 04:29 AM   #3
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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?
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April 3rd, 2018, 12:48 PM   #4
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The inequality comes from the definition of concave upward. You need to use a mathematical definition of this term.
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April 3rd, 2018, 06:51 PM   #5
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Quote:
Originally Posted by davedave View Post
Show that if f(x) is increasing and concave up on [a,b], then

f((a+b)/2) < (f(a) + f(a))/2.
This question makes no sense. There is nothing to show. This is literally the definition of concave up (usually called convex).

https://en.wikipedia.org/wiki/Convex_function
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April 5th, 2018, 10:00 AM   #6
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This question makes no sense. There is nothing to show. This is literally the definition of concave up (usually called convex).

https://en.wikipedia.org/wiki/Convex_function
Someone in my math club proved it. But, I don't remember how he did it.
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April 5th, 2018, 11:41 AM   #7
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Proved what? That every convex function is convex?
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April 5th, 2018, 01:50 PM   #8
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Quote:
Originally Posted by SDK View Post
Proved what? That every convex function is convex?
"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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