Question: $3$ Hessian in Optimisation problems $(15$ marks$)$

A three$-$product firm faces the following demand and cost functions:

$Q_1=90-6P_1-2P_2,Q_2=80-2P_1-4P_2,TC=2Q_1^2+3Q_1{Q}_2+2Q_2^2$

where $Q_1$ and $Q_2$ are complements, as indicated by the same sign for $P_1$ and $P_2$ in each

equation.

a$)(4$ marks$)$ Find the inverse demand functions using Cramer's rule

b$)(2$ marks$)$ Work out the profit function

c$)(3$ marks$)$ Find the output levels that satisfy the first$-$order condition for maximum

profit.

d$)(4$ marks$)$ Check the second$-$order sufficient condition. Can you conclude that

this problem possesses a unique maximum?

e$)(2$ marks$)$ What is the maximal profit?