Concave Function Properties

[Mark Cheung]

Given that u(w) is a concave function, and that u(w) ≥ 0.5u(w − 10) + 0.5u(w + 15) ,

1.) Prove, for any k≥1,
u(w+ 24 × k +1) − u(w+ 24 × k) ≤ 10 [u(w+ 24 × (k − 1) +1) − u(w+ 24 × (k − 1))] /15

2.) Hence show that for any fixed w, and any M > w, we have
u( w + M) − u( w ) < 72 [u( w + 1) − u( w)].

3.) Hence, prove
u(w) > 0.5u(w-72) + 0.5 u(w + M)

 

[Dr.Peterson]

Please show some work or ask specific questions, as the site asks you to. I can't tell what help you need, or what you are learning that you are expected to use here. (Why is this under Geometry/Trig?)

Also, are you using x as a variable, or to mean multiplication, which seems out of place here. That is, does u(w+ 24 × k +1) mean u(w+ 24k +1) or something else?

 

[Mark Cheung]

x is used to mean multiplication only, so for 1.), it's to prove u(w+ 24k +1) − u(w+ 24k) ≤ 10 [u(w+ 24(k − 1) +1) − u(w+ 24(k − 1))] /15,

It's actually a mathematical economics question about the properties of concave function, the ultimate goal is, give that
u(w) ≥ 0.5u(w − 10) + 0.5u(w + 15), u(w) is a concave function,
prove u(w) > 0.5u(w-72) + 0.5 u(w + M), for any M>0,

1.) and 2.) are intermediate steps.


I failed to prove the inequality in 1.), but i can use 1.) to prove 2.) (by putting k=1,2,3)
the thing that struggled me the most is how 1.) can be proved.. i tried to prove it by Mathematical induction, but i failed..

Regarding the category of this question, sorry that i put it in Geometry/Trig site, because i thought that concave function is a kind of geometry, maybe i should ask it in Algebra?

 

[Dr.Peterson]

As far as I'm concerned, the category doesn't matter; I just wanted to be sure of your context. Knowing that induction is the probable method (and that you know how to do that) is the important piece.

I don't have time at the moment, but I'll look into it in a while (or someone else will).