I am not sure if the IVT should be applied here. I am try to...

Jaxon Hamilton

Jaxon Hamilton

Answered

2022-07-23

I am not sure if the IVT should be applied here. I am try to do this problem and am stuck an how proceed:
Suppose f : [ 1 , 1 ] R is continuous and satisfies f ( 1 ) = f ( 1 ). Prove that there exists y [ 0 , 1 ] such that f ( y ) = f ( y 1 ).
So far, I have consideblack a new function g ( x ) = f ( x ) f ( x 1 ), x [ 0 , 1 ], but am stuck after this.

Answer & Explanation

renegadeo41u

renegadeo41u

Expert

2022-07-24Added 9 answers

Using the fact that f ( 1 ) = f ( 1 ), we have
g ( 0 ) = f ( 0 ) f ( 1 ) = f ( 0 ) f ( 1 )
g ( 1 ) = f ( 1 ) f ( 0 ) = [ f ( 0 ) f ( 1 ) ]
So, g ( 0 ) and g ( 1 ) have opposite signs. By continuity of g, it must have a root in [0,1], so there exists y [ 0 , 1 ] such that g ( y ) = 0, i.e. f ( y ) = f ( y 1 ).

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