1921] Akt Interesting Maximal Case 67 



and hence 



4bx . 



U = 4xy = Va- — x^ 



a 



since we reject here the consideration of a negative area, a l)eing 

 always greater than x. 



Let 



Then 



and 



f (x) = 



f(x) = xy/a- — x2 = (aV- — x4)M 



xCa'' — 2x^) _ a^ — 2x^ 

 (a-x2 — x4)3^ ~ (a2_x2)3^ 



(a'' — x2)J^ (— 4x) — (a^ — 2x2). ^x 



f"(x) = n — ;^^ ^ 



(a^ — x^) 

 _ x(2x'' — Sa") 



Setting f'(x) = 0, and solving we obtain 



a 



X = ± ^=- 

 V 2 



Since we have assumed that Va- — x^ is to have the positive sign, 

 in order to reject the consideration of negative areas, we obtain, on 



substituting x = + — =^in f"(x/ 

 V 2 



; a^ a^ 



\ 2 2 



a . 

 which proves that x = +'^^ gives a maxnnum value, as the con- 



dition for a maximum is that 



f"(A) = negative quantity. 



The same conclusion would have been reached had we expressed 

 U as an exphcit function of y, and followed a similar course of reason- 



