L24 Math 233 Lecture Notes - Lecture 24: Level Set, Minimax
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Math 233 lecture 24 applying lagrange"s idea. Lagrange"s idea (recap: minimize or maximize f(x,y) subject to constraint g(x,y) = k, draw curve g(x,y) = k. It follows that at p, f = g for some number : where the extremes occur, f = g for some number , solve for x, y, . G = <2x,2y: set up constraints/equations x2+y2 = 2 ey = 2 x xey = 2 y there are three unknowns and three equations, thus we can solve for all unknowns. Example 2 (three variables) find the extreme values of f(x,y,z) = xy2z, subject to g(x,y,z) = x2+y2+z2 = 4: calculate the gradients, extremes can only occur when f = g, solve for x, y, z, . G = <2x,2y,2z: set up constraints/equations x2+y2+z2 = 4 y2z = 2 x. = x2 y2z = 2 x y2 = 2x2 y = sqrt(2)x x2+y2+z2 = 4 x2+2x2+x2 = 4.
