The Pythagorean Theorem - Educational Outreach

Thus, the critical point ( C 2 ,0)is removed from furtherconsideration since F (C 2 ,0) 0 , which in turnA2 B2 C2implies. As a geometric digression, any point(x,0)on the lower segment of BndD represents adegenerate case associated with Figure 2.33 since a viabletriangle cannot be generated if the y value (the altitude) iszero as depicted in Figure 2.35A A12xy = 0C-xA 3Figure 2.35: Carolyn’s Cauliflower for y = 0xcp[ C xTo examine the locus of critical pointscp] y2 cp0, we first need to ask the followingquestion: Given a critical point x , y ) on the locus with(cp cp0 x cp C , does the critical point necessarily lie in IntD ?Again, we turn to the techniques of single-variabledifferential calculus for an answer.75