Intersections of two decreasing convex functions

Question

Suppose $f(x)$ and $g(x)$ are two continuous functions with the same domain and both are decreasing and convex. I think at most can intersect two times. Is this correct?

Answer 1

Quickie example. Domain is $(-\infty,0)$, functions are $$ f(x) = x^2-\sin(x),\qquad g(x) = x^2-\frac{\sin(2x)}{4} $$ Show: $f'(x)<0, f''(x)>0$ for all $x \in (-\infty,0)$. Intersect at least at all $\pi k$, where $k$ is a negative integer.