Show that f is convex on an interval

dgmath · #1 $Old$ November 17th, 2009, 06:12 PM

Hi there,
this is one of those problems I can't even figure out how to approach. Any idea would be highly appreciated.

Prove that

f is convex on an interval iff for all x and y in the interval,

f(tx + (1 − t)y) < tf(x) + (1 − t)f(y), for 0 < t < 1

tonio · #2 $Old$ November 17th, 2009, 08:58 PM

Quote:

Originally Posted by dgmath $View Post$

Hi there,
this is one of those problems I can't even figure out how to approach. Any idea would be highly appreciated.

Prove that

f is convex on an interval iff for all x and y in the interval,
f(tx + (1 − t)y) < tf(x) + (1 − t)f(y), for 0 < t < 1

As far as I am concerned this is exactly the definition of (upwards) convexity...which definition do you have?

Tonio

Drexel28 · #3 $Old$ November 17th, 2009, 08:59 PM

Quote:

Originally Posted by tonio $View Post$

As far as I am concerned this is exactly the definition of (upwards) convexity...which definition do you have?

Tonio
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I almost guarantee based on dgmath's previous posts that his definition is that $f''(x)>0$ . Of course, he/she missing some extra stipulations...differentiability etc.

dgmath · #4 $Old$ November 18th, 2009, 08:01 AM

Hi there,
I guess for the same reason, I'm stuck here. what I posted is pretty much all I have, but it IS true that it is a rephrased version of definition of convexity.