A firm is producing three goods and given inverse demand functions are

P1=140-4Q1-2Q2-2Q3, P2=240-2Q1-8Q2-4Q3, and P3=180-2Q1-2Q2-6Q3

And Total Cost =2Q1^2+4Q2^2+2Q3^2 +2Q1Q2 + 2Q1Q3 + 4Q2Q3. Now optimize the firm's profit for these three goods using Hessian

Method of optimization. First, find all the optimal values of Q1, Q2, Q3, and profit using

crammer's rule finally show that the profit is optimized (maximized) with these values.