The Pythagorean Theorem - Educational Outreach

Being polynomial in form, the function F is bothcontinuous and differentiable on D . Continuity impliesthat F achieves both an absolute maximum and absoluteminimum on D , which occur either on BndD or IntD .Additionally, F ( x,y) 0 for all points ( x,y)in D due tothe presence of the outermost square inF( x,y)y 4{ x[C x]2 } 2 .This implies in turn that F ( xmin, ymin) 0 for point(s)( xmin, ymin) corresponding to absolute minimum(s) forF on D . Equality to zero will be achieved if and only if2minx [ C x ] y 0 .minminReturning to the definition for F , one can immediately seethat the following four expressions are mutually equivalentF(xxAAmin22min, y22min[ C x B Bmin C) 0 ] y2 C22min 0 0 2 : We now employ the optimization methods ofmultivariable differential calculus to search for those points( xmin, ymin) where F ( xmin, ymin) 0 (if such points exist)and study the implications. First we examine F( x,y)forpoints ( x,y)restricted to the four line segments comprisingBndD .1.2F( x,0) 4{ x[C x]}. This implies F ( x,0) 0 onlyx or x C on the lower segment of BndD .when 0Both of these x values lead to degenerate cases.73