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If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being concave and then becomes convex from a point on, that g does not have to be convex, but can someone show me an example of an actual functional form that satisfies this property?

We know that since g has limit at infinity, g cannot be concave, but I am sure that there is a functional example of a function g:[0,∞)↦(0,∞) which is increasing, has limit zero at infinity, and is not everywhere convex, I just can't come up with it. Any ideas?

Thank you!

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    $g(x) = e^{-x^2}$ on $x \geq 0$.2012-07-07

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