Product of a concave function and a decreasing function

Let $f(x)$ and $g(x)$ be two positive continuous functions.
Function $f(x)$ is concave at $x_1$ and function $g(x)$ is $<1$ and is decreasing; $g(0)=1$ and $f(0)=0$.
Define function $h$ as the product of $f$ and $g$: $h(x)=f(x) g(x)$.

How can I prove (or disprove) that $h(x)$ increases from $0$ to reach a maximum at $x_2\le x_1$, and then decreases for $x > x_2$ ?

What happens if we replace $g$ by $g_1$, such that $g_1(x)<g(x)$ ? do we get $x_3 < x_2$ ? Note that in this case $h(x)=f(x)g_1(x)$ reaches its maximum at $x_3$..