Product of convex functions with special properties

Question

Let $f(x)$ and $g(x)$ be non-negative, convex functions in $C^2([M,\infty))$, where $M > 0$. Also, assume $f(x)$ is strictly decreasing on $[M,\infty)$, and that $g(x)$ is strictly increasing on $[M,\infty)$. We also have that $g(M) \geq f(M)$, and that $\lim_{x\to \infty} f(x)g(x) = 0$.

Does this imply that $f(x)g(x)$ is decreasing on the whole domain $[M,\infty)$?

Help much appreciated!

Edit: Added that $g(M) \geq f(M)$, and hence $g(x) > f(x)$ for all $x \in (M,\infty)$.

Answer 1

Try an example where $g(M) = 0$.

EDIT: For the revised question: $$ M = 1,\; f(x) = \frac{1}{x^2}, \; g(x) = \cases{x^3 & for $1 \le x \le 2$\cr 12 x - 16 & for $x \ge 2$} $$