I want to maximize a function of the form $f(x)=f_1(x)f_2(x)$ where $f_1(x)$ and $f_2(x)$ are differentiable and positive functions and are defined on the interval $(x_1,x_2)$. Further, I know the following about the functions
1- The function $f_1(x)$ strictly increases in the interval $x_1\leq x \leq a$ and it strictly decreases over the interval $a\leq x \leq x_2$.
2- The function $f_2(x)$ strictly increases in the interval $x_1\leq x \leq b$ and it strictly decreases over the interval $b \leq x \leq x_2$.
So if I want to maximize the function $f(x)$ then can I take the derivative of $f(x)$ and put it equal to zero?
My Current Understanding:
I think if the functions $f_1(x)$ and $f_2(x)$ are concave then the properties defined above are true for concave functions and we can take the derivative of $f(x)$ to get the maxima of the function $f(x)$. However, in my problem the functions $f_1(x)$ and $f_2(x)$ are not concave (since the second derivative condition does not hold). So I am confused about how to maximize the function $f(x)$? Any help in this regard will be highly appreciated. Thanks in advance.