1921] An Interesting Maximal Case 65 



Thus 



U = F(x,y) = 4xy . . . (1) 



subject to the condition 



X2 y2 



The critical values for maxima and minima are to be found by 



dv 

 equating to zero the first derivative of U, and replacing -p by its 



ClX 



specific value as derived from (2). 

 Thus 



dU dy 



-— = 4y + 4x -^ = from (l), 



dx dx 



and 



Hence 



ITTT 



which on substitution in the equation -p- = 0, 



dx 



gives 



— b^x\ 



— r) = 



a^y-' 

 and therefore 



a2y2 = b^X^ 



Combining with (2), we have 



2b2x2 = a2b2 



giving 



a , , . b 



X = ± • ■ / — and therefore y = ± y — ■ 



The criterion for a maximum value is 



-f— „ = negative quantity, 

 dx^ 



Now 



fi2TT dy dV dy 



— =4-7^ + 4x7^+4/- 

 dx2 dx dx2 dx 



