1. Chapter 14 – Partial Derivatives 14.8 Lagrange Multipliers 1 Objectives:  Use directional derivatives to locate maxima and minima of multivariable functions  Maximize the volume of a box without a lid if we have a fixed amount of cardboard to work with Dr. Erickson

2. Lagrange Multiplier Many optimization problems have restrictions or constraints on the values that can be used to produce the optimal solution. These constraints tend to complicate optimization problems because the optimal solution can occur at a boundary point of the domain. We use Lagrange Multipliers to simplify solutions. 14.8 Lagrange Multipliers2Dr. Erickson

3. Lagrange Multiplier λ is a Lagrange Multiplier 14.8 Lagrange Multipliers3Dr. Erickson

4. Method of Lagrange Multipliers 14.8 Lagrange Multipliers4Dr. Erickson

5. Visualization Lagrange Multipliers 14.8 Lagrange Multipliers5Dr. Erickson

6. Example 1 – pg. 963 #6 Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). 14.8 Lagrange Multipliers6Dr. Erickson

7. Example 2 – pg. 963 #10 Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s). 14.8 Lagrange Multipliers7Dr. Erickson

8. Example 3 Find the minimum value of the function subject to the constraint 14.8 Lagrange Multipliers8Dr. Erickson

9. Two Constraints The numbers λ and µ are the Lagrange multipliers such that 14.8 Lagrange Multipliers9Dr. Erickson

10. Example 4 – pg. 963 #16 Find the extreme values of f subject to both constraints. 14.8 Lagrange Multipliers10Dr. Erickson

11. Example 5 – pg. 964 #42 Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm 2 and whose total edge length is 200 cm. 14.8 Lagrange Multipliers11Dr. Erickson

12. Example 6 – pg. 964 #44 The plane intersects the cone in an ellipse. a)Use Lagrange multipliers to find the highest and lowest points on the ellipse. 14.8 Lagrange Multipliers12Dr. Erickson

13. More Examples The video examples below are from section 14.6 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 1 Example 1 ◦ Example 2 Example 2 ◦ Example 5 Example 5 14.8 Lagrange Multipliers13Dr. Erickson

14. Demonstrations Feel free to explore these demonstrations below. ◦ The Geometry of Lagrange Multipliers The Geometry of Lagrange Multipliers ◦ Constrained Optimization Constrained Optimization ◦ Visualizing the Gradient Vector Visualizing the Gradient Vector 14.8 Lagrange Multipliers14Dr. Erickson