206 MAXIMA. AND MINIMA VALUES 



equations F=Q, F l 0, F t = 0, will determine x, y, z, and u. 

 \\\> should then form the second differential coefficient of 



f(x, i/, z, u) with respect to x. This will involve -y-^ , % ^ 



a*u . 



and -, , , which must be found by differentiating equa- 



080 



tions (1) : by the sign of this second differential coefficient 

 off(x,y,z,u} we shall settle whether the function is a 

 maximum or a minimum. 



221. In Art. 214 we obtained as the condition for < (x) 

 having a maximum or minimum value, that </>'(#) must 

 change sign, and hence that $ (x) must be zero or infinite. 

 The cases in which <' (x) is infinite occur but rarely, and in 

 the Articles following Art. 214 we have always considered $' (x) 

 to vanish when </>(#) is a maximum or minimum. We shall 

 here add one proposition which shews that according to the first 

 view given of maxima and minima values (Arts. 209... 213), 

 a maximum or minimum may exist when the differential 

 coefficient of the function considered becomes infinite. 



Suppose that <(#) is such a function of x that when x = a 

 we have some of the differential coefficients of < (x) infinite, 

 so that <f> (a + k) cannot be expanded in powers of h by 

 Taylor's Theorem. 



Suppose that by some unexceptionable algebraical process 

 we find 



(a + h) - <j> (a) = Ah* + Eh* + GK 1 + ..., 

 where a, /3, 7, ..., are not necessarily positive integers. If 

 any one of these exponents be a fraction in its lowest terms 

 with an even denominator, then < (a h) (a) will be 

 impossible, and the consideration of maxima and minima 

 values becomes inapplicable. If none of the exponents be 

 of this form, then <f>(a h)<j> (a} will be a possible quantity. 

 Now there may be cases in which, by taking h small enough, 

 the sign of Ah a determines the sign of < (a + h) <J> (a) ; for 

 example, this happens if the number of terms in <f) (a+ h) <f> (a) 

 is finite, and the exponents or, y9, 7, ..., all positive, and a 

 the least. Let us suppose such a case, and let a be a proper 

 fraction with an even numerator ; then </> (a + h} </> (a) and 

 <f>(a h}^> (a) are both positive if A be positive, and nega- 

 tive if A be negative, when h is taken small enough. Hence 



