1. Optimization and Lagrangian

2. Partial Derivative Concept Consider a demand function dependent of both price and advertising Q = f(P,A) Analyzing a multivariate function often requires considering the independent variable impact on the dependent variable, all else equal. The partial derivative can be useful with this type of analysis. Consider the function

3. Optimization and Lagrangian Maximizing Multivariate Functions Maximize or Minimize functions by setting first order partial derivatives equal to zero. Again consider the function

4. Optimization and Lagrangian Maximizing Multivariate Functions in hundreds of dollars by substitution

5. Optimization and Lagrangian Role of Constraints (constrained optimization) subject to Solution cost with constraint

6. Optimization and Lagrangian Role of Constraints (constrained optimization) A positive second derivative is a minimum

7. Optimization and Lagrangian Lagrangian Multipliers (constrained optimization) Lagrangian multiplier incorporates the original objective function and the constraint conditions. written as

8. Optimization and Lagrangian Lagrangian Multipliers (constrained optimization) by subtraction multiplying by 7 then by adding which is and by substitution which is is then

9. Optimization and Lagrangian Lagrangian Multipliers (constrained optimization) Given it takes 4 fours of labor to produce output with only 300 hours available. which is is then