S4 On Maxima and Minima of Functions, fyc. 



d(ii) 



The equation — ^ =0 may, according to the cornmoa 



rules for maxima and minima, be thrown into the form 



K~) , '<v) 



— -T^ — ' = 0, or, if 71 be a fraction, and ==-, — =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 following 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. 



ROB. 



I. 



Having given the solidity of a Cone, to determine when 

 the curve surface is a minimum. 



In this case the height of the cone =x, and the radius of 

 the base =y. Let the latter be constant and =a. The 

 function v = l«ax, or varies as x, and u='jea'/a'' -{-x-, or 



/~ T G- 2 ^/a'+x '^ a'+x^- 



varies as V a- i-a;^. bmce n = |, , or 



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



found =aV2, whence -=V2. 

 y 



Prob. II. 



Having given the whole surface of a cone,, to determine 

 when the solidity is a maximum. 



In this problem, « varies as y/a-+x^ -i-a; and u is as a:. 



Also n = |, hence — ^ ,= max. which gives =2v^2. 



Va^^x^-{-a 



Prob. III. 



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

 ■^^hen the solidity is a maximum. 



