The Pythagorean Theorem - Educational Outreach

Letxcp(cpbe the x component of an arbitrary critical pointxcp, y ) . Now xcpmust be such that x cp Cx [ C x ] cp0 0 inorder for the quantity 0 . This in turn allowstwo real-number values foronly thosecpcpy values where y 0ycp. In light of Figure 2.33,cpare of interest.(x cp, C) & G(C) < 0G(y) isdefined onthis line(x cp, y cp)&G(y cp) = 0(x cp, 0) & G(0) > 0Figure 2.36: Behavior of G on IntDDefine the continuous quadratic function2 x [ C x ] on the vertical line segmentG( y)ycpconnecting the two points ,0)cp( cpx and ( x cp, C)on BndD asshown in Figure 2.36. On the lower segment, we haveG ( 0) xcp[C xcp] 0 for all xcpin 0 x cp C . On theupper segment, we have that G(C) x [ C x ] C2 0for allx in x cp Ccp0 . This is since the maximum thatG( x,C)can achieve for any xcpin 0 x cp C is 3C2 / 4per the optimization techniques of single-variable calculus.Now, by the intermediate value theorem, there must be avalue0 y* C where G(y*) x [ C x ] ( y*)2 076cpcpcp cp.By inspection, the associated point ( x cp, y*)is in IntD ,and, thus, by definition is part of the locus of critical pointswith x , y*) ( x , y ) .(cpcp cp