DIFFERENTIAL AND INTEGRAL CALCULUS. 87 



4. If /() = 1 and < (a) = co , and the limit of (x a) $> (x) , 

 when x = a, be m, then will the limit of {f(x}}* be e" 1 ''^. 



5. Define a maximum or minimum value of f(x) a 

 function of the independent variable #, and prove that 

 f(a) will be a maximum value of /(a?), if /' (#) change 

 sign from positive to negative as a? increases through the 

 value a ; and a minimum if f (x) change from negative 

 to positive. 



Find the maximum and minimum values of 



proving that the maximum value is 1, and the minimum 

 25 ; and explain how it happens that the minimum is 

 greater than the maximum. 



6. Find all the maximum and minimum values of 



- , x lying between and TT, and explain how it is 

 tan ox 



possible to haye, as appears in this case, two successive 



/ 3?r 57r\ 



minimum values [x = -, # = . 

 \ o o / 



dy 6 tan 3 ic cos 4.2 

 r = T-. Q x 2 , and changes sign not only when 



cix 

 smx and cos4# = 0, but also when cosx = 0). 



7. Prove that the expression x* has a maximum value 

 when x = s. 



8. A parabola is drawn having double contact with 

 a given circle ; prove that when the area included between 

 this parabola and the tangent to the circle perpendicular 

 to the axis of the parabola (DPAP'D 1 in fig. 17) is a 

 minimum, the latus rectum of the parabola is equal to 

 the radius of the circle. 



(If 4m be the latus rectum, a the radius of the circle, 



a 2 = CP*= 



