let f:[0,a]->RR of class C^1 such that f(0)=0 show that EEc in ]0,a[,f′(c)=(2f(a)+af′(a))/(3a)

Uroskopieulm

Uroskopieulm

Answered question

2022-11-17

let f : [ 0 , a ] R of class C 1 such that f ( 0 ) = 0 show that
c ] 0 , a [ , f ( c ) = 2 f ( a ) + a f ( a ) 3 a
First I apply mean value theorem then b ] 0 , a [ , f ( b ) = f ( a ) f ( 0 ) a 0 = f ( a ) a then f ( a ) = a f ( b )
so
2 f ( a ) + a f ( a ) 3 a = 2 3 f ( b ) + 1 3 f ( a )
How to prove that 2 3 f ( b ) + 1 3 f ( a ) is between f ( a ) and f ( b )?

Answer & Explanation

Claudia Woods

Claudia Woods

Beginner2022-11-18Added 15 answers

2 = 2 1 ;    4 = 2 2 ;    8 = 2 3 ;    16 = 2 4 ;       ; 2 n
akuzativo617

akuzativo617

Beginner2022-11-19Added 3 answers

If f ( b ) = f ( a ), it is easy. Otherwise, define
g ( x ) = f ( x ) [ 2 3 f ( b ) + 1 3 f ( a ) ] .
Then g ( x ) is continuous in [ b , a ] and
g ( b ) g ( a ) = 2 9 ( f ( b ) f ( a ) ) 2 < 0.
By the IMVT, there exists c ( b , a ) such that g ( c ) = 0 or
f ( c ) = 2 3 f ( b ) + 1 3 f ( a ) .

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