242 EXAMPLES OF MAXIMA AND MINIMA. 



than a + c. Therefore in passing from B to any adjacent 

 point either inside or outside the triangle the sum of the dis- 

 tances is increased; and therefore at the point B the sum is 

 a minimum. 



The values of -?- and -, - take the form - at the point 

 ax dy 



B ; and this is the reason that the solution failed to indicate 

 the point B. We have already remarked in Art. 234 that a 

 maximum or minimum value may exist corresponding to 

 such indeterminate values of the differential coefficients. 



6. Find the maximum and minimum value of 



(hx + ky a] (hx + ky V) 



Let u denote the expression, and let v denote 



then u = v' 1 (hx -\-ky-d) (hx + ky b)', 



du _h (Zhx + 2ky a I] 2x (hx + ky a) (hx + ky b) 

 dx v v* 



du _ k (2hx + 2ky a-b] _ 2y (hx + ky-a) (hx + ley - b) 

 dy~ v v* 



Put -5- = 0. and = : thus we deduce 

 dx dy 



Substitute rh for x and rk for y in -7- = or -=- = ; we 

 shall obtain after reduction the following quadratic equation 



n r : 



thus the values of r are possible, and one is positive and the 

 other is negative. 



