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.