Let f be continuous on [a,b] and differential on (a,b). Prove that if a >= 0 there are x1, x2, x3 ∈

ziphumulegn 2022-07-07 Answered
Let f be continuous on [a,b] and differential on (a,b). Prove that if a >= 0 there are x1, x2, x3 ∈ (a,b) such that
f ( x 1 ) = ( b + a ) f ( x 2 ) 2 x 2 = ( b 2 + b a + a 2 ) f ( x 3 ) 3 x 3 2
I think this problem will use the generalized mean value theorem to solve. However, I don't know how to apply it. Can you suggest a solution? Thank you very much!
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Answers (1)

iskakanjulc
Answered 2022-07-08 Author has 18 answers
By the mean value theorem,
(1) x 1 ( a , b ) : f ( b ) f ( a ) b a = f ( x 1 ) .
Consider g : [ a 2 , b 2 ] R defined by g ( x ) = f ( x ). Applying the mean value theorem to g, we infer that there is some t ( a 2 , b 2 ) such that
g ( b 2 ) g ( a 2 ) b 2 a 2 = g ( t ) ,
or equivalently, writing x 2 := t ,
(2) f ( b ) f ( a ) ( b a ) ( b + a ) = f ( t ) 2 t = f ( x 2 ) 2 x 2 .
Similarly, apply the mean value theorem to h : [ a 3 , b 3 ] R defined by h ( x ) = f ( x 1 / 3 ). We deduce that there is some s ( a 3 , b 3 ) such that
h ( b 3 ) h ( a 3 ) b 3 a 3 = h ( s ) ,
or equivalently, writing x 3 := s 1 / 3 ,
(3) f ( b ) f ( a ) ( b a ) ( b 2 + b a + a 2 ) = f ( s 1 / 3 ) 3 s 2 / 3 = f ( x 3 ) 3 x 3 2 .
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