Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics §7.5 LaGrangeMultipliers Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

7.4 Review § • Any QUESTIONS About • §7.4 → Least Squares Linear Regression • Any QUESTIONS About HomeWork • §7.4 → HW-07

§7.5 Learning Goals • Study the method of Lagrange multipliers as a procedure for locating points on a graph where constrained optimization can occur • Use the method of Lagrange multipliers in a number of applied problems including utility and allocation of resources • Discuss the signiﬁcance of the Lagrange multiplier λ

Lagrange Multipliers • Often the Domain of an Optimization is CONSTRAINED for some Reason; that is, • k a CONSTANT • The constraint Eqn could be solved for, say y: • In other words, the Constraint fcn describes a LINE in the xy-Plane Domain surface Constrained Domain LINE

Lagrange Multipliers Constrained Range LINE • The Constrained DOMAIN Line is then Projected Up or Down by the fcn • Functional projection produces a LINE on the Range Surface • It can be shown than any extremum on the range line must be a C.P. of

Lagrange Multipliers • Where λ is a new independent variable • To Find max/min for F(x,y) take • Solving the 3 eqns: • From the above equations determine the Critical Point (C.P.) Location: • Then

Lagrange Multiplier Method

Example  Lagrange Multipliers • Use the method of Lagrange multipliers to find the maximum value of • Subject to the Constraint of

Example  Lagrange Multipliers • SOLUTION • First find the partial derivatives of f & g: • And set each equal to the Lagrange multiplier, λ, times the partials of the left side of the constraint equation:

Example  Lagrange Multipliers • Solving the first two equations for λ: • By the Last Eqn: • Now use the Constraint Eqn: • The ONLY Soln to the last eqn:

Example  Lagrange Multipliers • Recall eqn for y(x): • Thus have Two Critical Points • Check max/min by functional evaluation • Thus the MAX value of 250 occurs at (5,−5)

Example  Find 2Var Domain • A seller’s assigned area is the six-mile radius surrounding the center of a city. • History indicates that x miles east and y miles north of city center, his/her sales competition by other businesses is Modeled by • Find • the location(s) for minimum competition • The minimum level of competition

Example  Find 2Var Domain • SOLUTION • The constraint for this function is the circle of radius six miles centered about the middle of the city. Such a circle can be described by the points (x,y) satisfying the equation: • Taking the partials of the competition function find:

Example  Find 2Var Domain • In this case g(x,y) = k → • ReCall the Lagrange Equation: • Then the Lagrange Multiplier Minimum System

Example  Find 2Var Domain • Using eqn (1) to Solve for y • To prevent Division by ZeroSpecify x ≠ 0 • Use the above result in eqn (2) • SolvingtheAbove

Example  Find 2Var Domain • Combining this result with the solution for y in terms of λ and the constraint equation to solve for λ:

Example  Find 2Var Domain • Finally, use the value of λ to determine values of x & y for minimum competition: • Testing the Four (x,y) Pairs find: • Thus the minimum of 1.69 businesses occurs 3.46 miles north and 4.90 miles either east/west of the center of the city

Lagrange Multiplier as a Rate • Thus λ is a Marginal Rate for the max or min with respect to a change in the constraint value

Example  Lagrange as Rate • In the Previous the minimum value was M=1.69 Businesses, with k = 36 sq-miles • If k increased by 1 sq-mi (in context this would be increasing the radius of the seller’s route), the approximate change in the minimum value: • The min no. of competing businesses would INcrease by about 0.346

WhiteBoard Work • Problems From §7.5 • P7.5-32 →ConstantElasticityof Substitution(CES)ProductionFunction

All Done for Today • Born: 25 January 1736 • Died: 10 April 1813 (aged 77) • Professorship • ÉcolePolytechnique • Academic advisors • Leonhard Euler • Giovanni Beccaria • Doctoral students • Joseph Fourier • Giovanni Plana • Siméon Poisson JosephLouisLagrange

Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

Q := 50*(0.3*K^(-1/5) + 0.7*L^(-1/5))^-5 dQdK = diff(Q, K) dQdL = diff(Q, L) K := 140/(5+2*(35/6)^(5/6)) Kn := float(K) L := K*(35/6)^(5/6) Ln := float(L)

Qmax = subs(Q, K = Kn, L = Ln) Qmax = subs(Q, K = K, L = L)

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege