I consider the cost function given by $$ TC=q_1 +kq_2 $$ $ k\in(0,1). $ Demand functions are given by $
$$ q_1(p_1,p_2)=q_2(p_1,p_2)=(p_1p_2)^{-3}, p_1>0, p_2>0 $$ I need to find optimal values for prices and find the values of k for which one of the products is priced under marginal cost.
The answer is $$ p_1=p_2=3/5(1+k), k>3/2 $$ When I rewrite TC as $$ TC=(k+1){{(p_1p_2)}^{-3}} $$ I find the FOC(first-order conditions): $$ \begin{equation} \frac{\partial TC}{\partial p_1}, \frac{\partial TC}{\partial p_2} \end{equation}$$ which is unhelpful for finding critical points $p_1, p_2$ and that's why I can't proceed to finding definiteness of Hessian. Thanks for Your time and consideration.
Hint: The optimal prices are those which maximise total profit, so the first-order optimality conditions are not $$ \frac{\partial TC}{\partial p_1}= \frac{\partial TC}{\partial p_2}=0\ , $$ but $$ \frac{\partial TP}{\partial p_1}= \frac{\partial TP}{\partial p_2}=0\ , $$ where \begin{eqnarray} TP &=& p_1q_1 + p_2q_2 - q_1 - kq_2\\ &=& \frac{1}{p_1^2p_2^3} + \frac{1}{p_1^3p_2^2}-\frac{1+k}{p_1^3p_2^3} \end{eqnarray}