I am trying to maximize $P = 20q_a - 2q_a^2 + 16q_b-2q_b^2 – 1/4 (q_a + q_b)^2$, a profit equation, function of two separate quantities of products. I thought that what I need to do is to set both derivatives =0 and find where that equation holds for both variables, but I am having some trouble. Here's my work:
$dP/dq_a = 20 – 4q_a – 1/2 (q_a + q_b) = 0$
$dP/dq_b = 16 – 4q_b – 1/2 (q_a + q_b) = 0$
$4 – 4q_a +4q_b = 0$
$4 = 4 (q_a – q_b)$
$q_a = q_b + 1.$
As you can see from the last equation, I get $q_a = q_b + 1$, but when I plug a pair of 1, 0 ($q_a$ and $q_b$) into the derivative equations, I do not get 0's, so I guess I made a mistake somewhere (wolfram alpha gives a definitive answer). Please help, thanks!