49 show that the necessary and sufficient condition

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* 49. Show that the necessary and sufficient condition that Z f ( x ) { F ( x ) } 2 dx, where f and F are polynomials of which the latter has no repeated factor, should be a rational function of x , is that f 0 F 0 - fF 00 should be divisible by F . ( Math. Trip. 1910.) 50. Show that Z α cos x + β sin x + γ (1 - e cos x ) 2 dx is a rational function of cos x and sin x if and only if αe + γ = 0; and determine the integral when this condition is satisfied. ( Math. Trip. 1910.) * See the author’s tract quoted on p. 286 .
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CHAPTER VII ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND INTEGRAL CALCULUS 147. Higher Mean Value Theorems. In the preceding chapter ( § 125 ) we proved that if f ( x ) has a derivative f 0 ( x ) throughout the interval [ a, b ] then f ( b ) - f ( a ) = ( b - a ) f 0 ( ξ ) , where a < ξ < b ; or that, if f ( x ) has a derivative throughout [ a, a + h ], then f ( a + h ) - f ( a ) = hf 0 ( a + θ 1 h ) , (1) where 0 < θ 1 < 1. This we proved by considering the function f ( b ) - f ( x ) - b - x b - a { f ( b ) - f ( a ) } which vanishes when x = a and when x = b . Let us now suppose that f ( x ) has also a second derivative f 00 ( x ) throughout [ a, b ], an assumption which of course involves the continuity of the first derivative f 0 ( x ), and consider the function f ( b ) - f ( x ) - ( b - x ) f 0 ( x ) - b - x b - a 2 { f ( b ) - f ( a ) - ( b - a ) f 0 ( a ) } . This function also vanishes when x = a and when x = b ; and its derivative is 2( b - x ) ( b - a ) 2 { f ( b ) - f ( a ) - ( b - a ) f 0 ( a ) - 1 2 ( b - a ) 2 f 00 ( x ) } , and this must vanish ( § 121 ) for some value of x between a and b (exclusive of a and b ). Hence there is a value ξ of x , between a and b , and therefore capable of representation in the form a + θ 2 ( b - a ), where 0 < θ 2 < 1, for which f ( b ) = f ( a ) + ( b - a ) f 0 ( a ) + 1 2 ( b - a ) 2 f 00 ( ξ ) . 319
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[VII : 147] ADDITIONAL THEOREMS IN THE CALCULUS 320 If we put b = a + h we obtain the equation f ( a + h ) = f ( a ) + hf 0 ( a ) + 1 2 h 2 f 00 ( a + θ 2 h ) , (2) which is the standard form of what may be called the Mean Value Theorem of the second order . The analogy suggested by (1) and (2) at once leads us to formulate the following theorem: Taylor’s or the General Mean Value Theorem. If f ( x ) is a function of x which has derivatives of the first n orders throughout the interval [ a, b ] , then f ( b ) = f ( a ) + ( b - a ) f 0 ( a ) + ( b - a ) 2 2! f 00 ( a ) + . . . + ( b - a ) n - 1 ( n - 1)! f ( n - 1) ( a ) + ( b - a ) n n ! f ( n ) ( ξ ) , where a < ξ < b ; and if b = a + h then f ( a + h ) = f ( a ) + hf 0 ( a ) + 1 2 h 2 f 00 ( a ) + . . . + h n - 1 ( n - 1)! f ( n - 1) ( a ) + h n n ! f ( n ) ( a + θ n h ) , where 0 < θ n < 1 . The proof proceeds on precisely the same lines as were adopted before in the special cases in which n = 1 and n = 2. We consider the function F n ( x ) - b - x b - a n F n ( a ) , where F n ( x ) = f ( b ) - f ( x ) - ( b - x ) f 0 ( x ) - ( b - x ) 2 2! f 00 ( x ) - . . . - ( b - x ) n - 1 ( n - 1)! f ( n - 1) ( x ) .
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[VII : 147] ADDITIONAL THEOREMS IN THE CALCULUS 321 This function vanishes for x = a and x = b ; its derivative is n ( b - x ) n - 1 ( b - a ) n F n ( a ) - ( b - a ) n n ! f ( n ) ( x ) ; and there must be some value of x between a and b for which the derivative vanishes. This leads at once to the desired result. In view of the great importance of this theorem we shall give at the end of this chapter another proof, not essentially distinct from that given above, but different in form and depending on the method of integration by parts.
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