Let's assume that f:[0,∞[ -> R is differentiable twice and f(0)=0 and f''(x)>0, when x>0.

Show that f(x)+f(y)<f(x+y) when x,y>0.

It's really important that I get this done, so any help is highly appreciated!!

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- May 12th 2010, 03:24 AM #1

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- May 12th 2010, 10:52 AM #2

- May 14th 2010, 12:56 AM #3

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- May 14th 2010, 01:46 AM #4
The only reason that you need the condition f''(x)>0 (for x>0) is that it implies that f'(x) is a strictly increasing function, so that f'(x+y) > f'(x) when y>0.

The value of f''(x) when x=0 is not relevant. The fact that f''(x) > 0 when x>0 does in fact imply that f is strictly convex.