204 MAXIMA AND MINIMA VALUES 



Substitute the value of y in v = 0, and we have 



a \ a 



therefore x = a + 



- 



Upon examination it will be found, that if we take the 

 upper sign in the value of x we obtain a maximum value 

 for u, and if we take the lower sign, a minimum. This 

 example is a solution of the geometrical question, " To find 

 the points in the circumference of a given circle which are at 

 a maximum or minimum distance from a given point." 



219. The process for finding the maxima and minima 

 values of an implicit function may be extended to the case 

 in which one variable is connected with more than one other 

 variable, the whole number of equations being one less than 

 the whole number of variables. Suppose, for example, we 

 have three equations, 



F(x, y, z, w)=0, 

 FJx, y, z, u} = 0, 



u being the variable of which we wish to find the maximum 

 or minimum value. 



From the given equations it follows that we may consider 

 y, z, and u functions of the independent variable x. Hence 



dj^ dF dy ^'^,^^ = 



dx dy dx dz dx du dx 



df\ dF\dy_ df\ dz_ dF t du 



dx dy dx dz dx du dx 



dF, df\dy_ dF s dz dF,du =Q 



dx dy dx dz dx du dx 



.(1). 



From these equations we can eliminate -f- and -y- , and 



du *" dx 



the value of -- which we then obtain must be put equal 



