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?