Suppose that f(0)<f(1). Consider now, f(0)<sqrt(c)<f(1). By the intermediate value theorem, EEb in [0,1]:f(b)=sqrt(c). Now we need to show that its possible that b=c, but this is exactly where I am stuck.

Raina Gomez 2022-09-22 Answered
Use the intermediate value theorem to prove that if f : [ 0 , 1 ] [ 0 , 1 ] is continuous, then there exists c [ 0 , 1 ] such that f ( c ) = c
Suppose that f ( 0 ) < f ( 1 ). Consider now, f ( 0 ) < c < f ( 1 ). By the intermediate value theorem, b [ 0 , 1 ] : f ( b ) = c .
Now we need to show that its possible that b = c, how?
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Answers (1)

madleeinluvec
Answered 2022-09-23 Author has 5 answers
Write g ( x ) = f ( x ) x , g ( 0 ) 0 , g ( 1 ) 0, so there exists c such that g ( c ) = f ( c ) c = 0.
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