Optimize the profit function using Hessian.

The firm is producing three goods chairs (C), table (t), and sofa (S) with processed woods.

Demand Function for chair C=200-6Pc+2Pt-2Ps,
Demand Function for table T=360+2PC-8Pt+4Ps and
Demand Function for sofa S=400+4Pc+2Pt-10Ps where
Total cost TC=2C² + 47² +25² + 2CT + 27S + 2CS

Optimize the profit function using Hessian.
Where C, T, and S represents the quantity of chair, table and sofa, and Pc, Pt, Ps represents the respective price of them.