If the values of both functions are nonnegative, then the product of minima is less than or equal to the minimum of products. This relies on the following fact about real numbers: if $0\le x_1\le y_1$ and $0\le x_2\le y_2$, then $x_1x_2\le y_1y_2$.
For example, consider functions $f$ and $g$ and just two values in the domain: $x_1$ and $x_2$. Let $a_1=f(x_1)$, $a_2=f(x_2)$, $b_1=g(x_1)$, $b_2=g(x_2)$, Then $\min(a_1,a_2)\le a_1$ and $\min(b_1,b_2)\le b_1$, so by the fact above $\min(a_1,a_2)\min(b_1,b_2)\le a_1b_1$. Similarly $\min(a_1,a_2)\min(b_1,b_2)\le a_2b_2$, so $\min(a_1,a_2)\min(b_1,b_2)\le\min(a_1b_1,a_2b_2)$.
If the numbers can be negative, then this conclusion no longer holds. For example, if $a_1=1$, $a_2=2$ and $b_1=b_2=-1$, then $\min(a_1,a_2)\min(b_1,b_2)=1\cdot(-1)=-1>-2=\min(-1,-2)=\min(a_1b_1,a_2b_2)$,