L24 Math 233 Lecture Notes - Lecture 23: 32X, Level Set
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Math 233 lecture 23 extreme values. {(x,y) | x2+y2<1}: not closed, missing its entire boundary. {(x,y) | x2+y2 1, but (x,y) (1,0)}: not closed, missing one boundary point. Find the extreme values of f(x,y) = x+y-xy on the triangle with vertices (0,0), (0,2) and (4,0) C is the segment y = -1/2x+2, 0 x 4. Critical points fx = 1-y fy = 1-x. Segment a f(x,y) = f(x,0) = g(x) = x g minimum = g(0) = 0 = f(0,0) g maximum = g(4) = 4 = f(4,0) Segment b f(x,y) = f(0,y) = h(y) = y h minimum = h(0) = 0 = f(0,0) h maximum = h(2) = 2 = f(0,2) Compare all values at critical points and boundary, we can conclude. Lagrange"s idea: minimize or maximize f(x,y) subject to constraint g(x,y) = k, draw curve g(x,y) = k.
