If a function is strictly decreasing and convex, what can we say about its first derivative?

Question

Suppose we have a strictly decreasing convex differentiable one-variable function $f(x)$, $x\in (0,\infty)$. Can we infer from this that its derivative will always be convex/concave? For example, if $f(x)=1/x$, then $f'(x)=-1/x^2$. In this case $f'(x)$ is concave, but I am not sure that $f$ strictly decreasing and convex $\rightarrow$$f'$ concave/convex always holds. Thanks.

Answer 1

Take $f(x)=-x+(x-1)^4/4, 0<x<2\\f'(x)=-1+(x-1)^3\\f''(x)=3(x-1)^2$

Then $f'<0,f''\ge0$ indicating that $f$ is strictly decreasing and convex, yet $f'$ is neither convex nor concave.

Graph of $f(x)$ Graph of $f'(x)$