2
$\begingroup$

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?

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

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.