14.8+note+outline.pdf - 14.8 Lagrange Multipliers In this section we learn Lagranges method for maximizing or minimizing a general function f x y

14.8+note+outline.pdf - 14.8 Lagrange Multipliers In this...

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1 14.8 Lagrange Multipliers In this section, we learn Lagrange’s method for maximizing or minimizing a general function ( ,)f x ysubject to a constraint of the form ( ,)g x ykWe seek the extreme values of fwhen the point is restricted to lie on the level curve ( ,)g x yk(for functions of two independent variables) or on the level surface ( , , )g x y zk(for functions of three independent variables). Consider the functions of two independent variables case: To maximize ( ,)f x ysubject to ( ,)g x ykis to 1.According to Section 14.7, 2.But there is another way to look at this. According to Section 14.8, consider the projection of this problem onto the xy-plane. This happens when the two level curves The normal lines (lines passing through a point Pthat are perpendicular to the tangent line at that point) are The gradient vectors ,fgare Method of Lagrange Multipliers To find the maximum and minimum values of ( ,)f x ysubject to the constraint ( ,)g x yk(assuming that these extreme values exist and that 0g .
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