math16B_suppl_notes_3

math16B_suppl_notes_3 - Math 16B – S06 – Supplementary...

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Unformatted text preview: Math 16B – S06 – Supplementary Notes 3 The Lagrange Multiplier Method The method of Lagrange multipliers is often effective in finding solutions of constrained ex- tremum problems. In the two-variable version of such a problem, one is given a function f ( x, y ), and one wishes to maximize it or minimize it under the constraint that another function g ( x, y ) vanishes (i.e., one wishes to find a maximum or minimum of f on the level curve g ( x, y ) = 0). As explained in our textbook (where you will also find examples), Lagrange’s method proceeds as follows. One introduces a third variable λ (traditionally called a Lagrange multiplier), and one defines a function F ( x, y, λ ) of three variables by F ( x, y, λ ) = f ( x, y ) + λg ( x, y ) . The basic theorem underlying the method states that if f ( x, y ) attains a maximum or a minimum at the point ( a, b ) under the constraint g ( x, y ) = 0, then there is a value c of λ such that ( a, b, c ) is a critical point of F : (1) ∂F...
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math16B_suppl_notes_3 - Math 16B – S06 – Supplementary...

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