The Pythagorean Theorem - Educational Outreach

2 22. F(x,C) 4{ x[C x] C } 0 for all points on theupper segment of BndD since the smallest value2that | x[C x] C | achieves is 3C 2 / 4 , asdetermined via the techniques of single-variabledifferential calculus.3. F (0, y) F(C,y) 4y4 0 for all points other thany 0 (a degenerate case) on the two verticalsegments of BndD .To examine F on IntD , first take the partial derivatives ofF with respect to x and y . This gives after simplification:F(x,y) / x 8[ C 2x]{x[C x]F(x,y) / y 16y{x[C x]2y}2yNext, set the two partial derivatives equal to zF ( x,y) / x F(x,y) / y 0 .Solving for the associated critical points x , y ) yields}(cp cpone specific critical point and an entire locus of criticalpoints as follows:(C 2 ,0) , a specific critical point2cpx [ C x ] y 0 ,cpcpan entire locus of critical points.The specific critical point (C 2 ,0)is the midpoint of the lowersegment for BndD . We have that4F(C ,0) C / 4 0 .2 74