84 On Maxima and Minima of Fimttions^ f^'C. 



r » 



(^) 



ooni 



The equation — ^ 

 rules for maxima and minima, be thrown into the form 



— 0, or, if n be a fraction, and =-, — i =0. The 



same is true of the equation — ^ =o. Either of these 



forms may be used in individual cases, as is most conven- 

 ient. 



The follovving problems, which chiefly respect isoperi- 

 meters, will be sufficient to exemplify, and to shew the ad- 

 vantages of this method. To avoid confusion, the variable 

 quantity x or y which is considered constant, will be put^ 

 during the operation, =a. 



r^ 



PROB, I. ; 



Having given the solidity of a cone, to determine when 

 the curve surface is a minimum. 



. In this case the heiglit of the cone =0^, and the radius of 

 the base =y. Let the latter be constant and =flr- The 



function v^l'jfax^ or varies as 3:, and u^'^ay/a^ -{-x^^ or 



varies as Va^ i-j:^. Since n = |, z , or 





^1 ' xf 



min. Making the ditTerential of this fraction =0, x is 



found =aV2. whence ^=V2. 



-■* 



Peob. II 



s*,.* 



Having given the whole surface of a cone, to determine 

 «^hen the solidity is a maximum. 



In this problem, v varies as %^a- -fa;' +a ; and tt is as a: 



Also n = |, hence — ^ ^ ^ = max, which gives =2i/2 



/a^+x^ + a ^ 



Pkob. 1£L 



The whole surface of a regular prism being given, to find 

 when the solidity is a maximum. 



