EXAMPLES VIII 127 



when t " b + y = ae~ k ( b+ p' sin a 

 P 



M 



Thus the successive maximum values of y are in geometrical 

 progression, the common ratio of which is e~~P~ 



Also when t = b+ - y = - ae~ k ^ + pj sin a = N 



when t=b+~ y ' = ae~ k ( b+ ~p) sin a 



= Ne ~p~ 

 when t = b+- y = - ae~\ + ~p) sin a 



= Ne T" 

 Thus the successive minimum values of y are in geometrical 



progression, the common ratio of which is e~~p~ 



EXAMPLES VIII 



Find the maximum and minimum values of y and the values 

 of x, producing them in each of the following examples: 



(1) y = 2X 3 - 9x 2 + I2x + 30. 



(2) y = x 3 - 75x + 24. 



(3) y = 3x* + 4a^ 24a; 2 4&r + 64. 



\ / V 



Q 



(4) Find the minimum value of - + 5x 3 , and the value of x 



x 



which produces it. 



In each of the following examples find the maximum value of 

 x, and the value of t which produces it. 



(5) x=Se~ t sin St. 



(6) x = \/Io e-* sin (3t + 1-249). 



(7) x = 5e-* sin (Si + 0-6428). 



(8) x = lOt e-v. 



(9) x = 5 (1 + 2t) e~*. 



(10) x = 5 (1 + 4<)e- 2 '. 



(11) x= 4 (e-* -e-"}. 



(12) x = 4 (2e~* - e-*<). 



(13) x= 4 (3e-*- 2-*). 



(14) If y = a^e"*, find the maximum value of y, and the value 

 of x producing it. 



(15) Find the dimensions of the cylinder of greatest volume 

 /hich can be inscribed in a sphere of 10 inches radius. 



