Lecture 13 Lagrange Multiplier Method Finding maximums and minimums with constraints Example Maximize f x y xy with the constraint g x y x y 1 0 So we need to nd points where level curves of f and g touch but do not cross This means they have the same normal direction f normal to f constant g normal to g constant In original problem Square and add the rst two equations An easier way in this case g 0 is parameterized by cost sint for 0 t 2 w t f cost sint cost sint w sin t cos t w 0 when x y this is exactly what we got before Legrange multiplier theorem If the maximum or minimum of f such that g 0 occurs at a point P where g P 0 then f P g P for some Example f x y z nd the maximum g x y z 3 0 Take curve r t in g 0 r t g If we are at a point that is a maximum or minimum for f on g 0 its also a maximum or minimum on the curve More constraints maximize f x y z such that g x y z 0 and g x y z 0 These level curves could also be found in the third quadrant to minimize this function we should look in the second and fourth quadrants
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