Talk:Tangent Line to Convex Graph

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Is the converse to this theorem true as well, that the tangent line being below the graph implies $f$ is concave up? If so, I can reword the theorem to make it a stronger statement. --GFauxPas 16:51, 27 November 2011 (CST)

In fact, a more general definition of a concave up function is:

A function (not necessarily continuous) is said to be concave up on an interval $I$ iff:
$\forall x,y,z\in I: x<y<z \implies f(y) > f(x) + \dfrac{y-x}{z-x}(f(z)-f(x))$
Similar definitions exist for concave down and convex up/down. My resource is the book 'Measures, Integrals and Martingales' by R. Schilling.
So I am not sure what your definition of concave up is (probably something weaker with a twice differentiable function and a condition on the second derivative). --Lord_Farin 13:25, 28 November 2011 (CST)
So I have to find a different terminology then. Larson defines a real differentiable function $f$ to be concave up on $\mathbb I$ iff $\forall x_1,x_2 \in \mathbb I, x_1 > x_2 \implies f'(x_1) > f'(x_2)$. He also doesn't use the term convex at all, just concave up and concave down. Khan Academy uses the same definition. Suggestions? --GFauxPas 13:36, 28 November 2011 (CST)
I can not give a name to the property and just say that $f'$ is strictly increasing on $\mathbb I$.? --GFauxPas 14:34, 28 November 2011 (CST)
It is clear that my definition encapsulates yours (well, if not, convince yourself; it will require proof in the future anyway, and this will fit in somewhere I'm sure) so I suggest the definition to be done in your way first (for concave up/down), and then I will generalise it later on, with proof that it is a generalisation; I will then also add alternative terminology (which is usually concave vs. convex instead of concave down/up (IIRC (whoa, I like nested parentheses :) )). --Lord_Farin 14:46, 28 November 2011 (CST)