Let $f(x)$ be a twice-differentiable function. The definitions for concave upward and concave downward I found in my textbooks are all somewhat wordy, something along the lines of:

Definition 1: $f(x)$ is concave upward at $c$ if $f(x) > [f(c)+f(c)(x−c)]$ for all $x$ in some open interval containing $c$.

(See here for example.)

Why don't we simply use the following definition:

Proposed Definition 2: $f(x)$ is concave upward at $c$ if $f''(c) > 0 $

Are there cases where $f''(c) > 0$, but Definition 1 is not satisfied? Aren't Definitions 1 and 2 equivalent?

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    $\begingroup$ Definition 1 is more general because not all functions are twice-differentiable. $\endgroup$ – Christopher A. Wong Oct 30 '15 at 2:31

You can try to prove yourself the equivalence between the two assertions in the case of the function $f$ is two time differentiable, it is quite easy.

$f$ two time diffenrentiable, $f(x) > [f(c)+(x-c)f(c)] \Rightarrow f'(x) > f(x) \Rightarrow f''(x) > 0$ and the other way...

But consider that a function can be not two time differentiable yet still convexe. The first definition is more general.


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