Lagrange Multipliers Academic Resource Center In This Presentation..

•We will give a definition •Discuss some of the lagrange multipliers •Learn how to use it •Do example problems Definition Lagrange method is used for maximizing or minimizing a general function f(x,y,z) subject to a constraint (or side condition) of the form g(x,y,z) =k. Assumptions made: the extreme values exist ∇g≠0 Then there is a number λ such that ∇ f(x0,y0,z0) =λ ∇ g(x0,y0,z0) and λ is called the Lagrange multiplier. • Finding all values of x,y,z and λ such that ∇ f(x,y,z) =λ ∇ g(x,y,z) and g(x,y,z) =k And then evaluating f at all the points, the values obtained are studied. The largest of these values is the maximum value of f; the smallest is the minimum value of f. • Writing the vector equation ∇f = λ ∇g in terms of its components, give ∇fx = λ ∇gx ∇fy = λ ∇gy ∇fz = λ ∇gz g(x,y,z) =k • It is a system of four equations in the four unknowns, however it is not necessary to find explicit values for λ. • A similar analysis is used for functions of two variables