Examples of $C^1$ differentiable convex functions.

Question

Could you please provide examples of convex functions that are differentiable, but their derivatives are not differentiable.

Here is a nice result for building such functions: http://math.stackexchange.com/questions/143721/about-convex-function — charlestoncrabb, Nov 16 '15 at 20:43

score 1 · Accepted Answer · answered Nov 16 '15 at 20:42

Define $f$ as follows:

$$f(x)= \begin{cases} & 2x^2 \text{ if } x \leq 0 \\ & 3x^2 \text{ if } x>0 \end{cases}$$

Then $f$ is differentiable with:

$$f'(x)= \begin{cases} & 4x \text{ if } x \leq 0 \\ & 6x \text{ if } x>0 \end{cases}$$

$f'$ is monotonically increasing, thus $f$ is convex. Yet the second derivative does not exist at $x=0$.

score 1 · Answer 2 · answered Nov 16 '15 at 20:43

1

$f(x)=\left\{\begin{matrix} x^2, x\geq 0\\ 0, x< 0 \end{matrix}\right.$

answered Nov 16 '15 at 20:43

Erfan Khaniki

2 Answers2