AN INTERESTING MAXIMAL CASE* 

 By Archibald Henderson and H. G. Baity 

 Plate 3 



In lists of problems dealing with maxima and minima may com- 

 monly be encountered the following: 



To find the dimensions of the rectangle of maximum area that can 

 be inscribed in an ellipse of semi-axes a and 6. 



Various methods may be employed for the solution of this prob- 

 lem; but these ordinarily suffice to veil the characteristic features 

 of the solution, as viewed in the present paper. An enumeration of 

 the methods usually employed will bring this into prominence. The 

 interesting feature of the problem, under the method here stressed, 

 is the introduction of an auxiliary curve which plays a dominant role 

 in the solution. 



Method I 



If we represent a critical value by A and Ax by h, the customary 

 conditions for a maximum are 



f (A — h) — f (a) = positive quantity 



f (A) — f (A + h) = positive quantity 

 If we represent the sides of the rectangle, inscribed in the ellipse 



X^ y2 



— + — = 1 by 2x and 2y, the area of the rectangle is given by 



U = 4xy. 



Since, from the equation of the elUpse, y = - 1 a^ — x^, 



a 



a a 



Let 



f (x) = (a^x^— x*)3^ = x(a2— x2)i^. 



To find the critical value, set f'(x) = 



.' . V2 (a^xs — x^) --M (2a2x — 4x') = 0. 

 Hence 



2x {a? — 2x2) = 0; 



which gives x = 0, ± —-=, 



* This investigation, which had been directed by Dr. Henderson, was presented by 

 Mr. Baity before the Mathematics Club, University of North Carolina. 



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