Estimates maximum of linear equation division

Question

I have a really hard and confusing problem like below:

$f(A,B)=\frac{k_{2_{A_2}}*k_{2_{A_2}}*A_2+k_{1_{B_2}}*k_{2_{B_2}}*B_2}{k_{1_{A_1}}*k_{2_{A_1}}*A_1+k_{1_{B_1}}*k_{2_{B_1}}*B_1}$

Then I define like this:

$f_A = \frac{k_{1_{A_2}}}{k_{1_{A_1}}} * \frac{k_{2_{A_2}}}{k_{2_{A_1}}} * \frac{A_2}{A_1}$

$f_B = \frac{k_{1_{B_2}}}{k_{1_{B_1}}} * \frac{k_{2_{B_2}}}{k_{2_{B_1}}} * \frac{B_2}{B_1}$

Then $f(A,B)$ can be simplified as:

$f(A,B)=\frac{f_A*k_{1_{A_1}}*k_{2_{A_1}}*A_1+f_B*k_{1_{B_1}}*k_{2_{B_1}}*B_1}{k_{1_{A_1}}*k_{2_{A_1}}*A_1+k_{1_{B_1}}*k_{2_{B_1}}*B_1}$

What I have are:

1. The value of $A_1$ and $B_1$
2. $\frac{A_2}{A_1}$ and $\frac{B_2}{B_1}$
3. The ratio of $k_1$ is between 0 and 1
4. The ratio of $k_2$ is unknown but I think it can be assumed that the maximum of $k_2$ is equal to the ratio of $\frac{A_2}{A_1}$ and $\frac{B_2}{B_1}$

Is it possible to estimate the maximum value of $f(x)$ with regard to $f_A$ and $f_B$ (a function with only $f_A$ and $f_B$ as parameter)?

If not possible, what assumption do I need to make so that it is possible?

Thank you.